departamento de matematicas´math0.bnu.edu.cn/~zhengc/material/thesis.pdf · 2015. 3. 9. ·...

DEPARTAMENTO DE MATEMATICAS

FACULTAD DE CIENCIAS

UNIVERSIDAD AUTONOMA DE MADRID

CONTROL OF TIME-DISCRETE

APPROXIMATION SCHEMES FOR

PARTIAL DIFFERENTIAL EQUATIONS

Memoria para optar al tıtulo de Doctor en Ciencias Matematicas

Presentada por

Chuang Zheng

Dirigida por

Enrique Zuazua Iriondo

Madrid, 2007

Agradecimientos

Durante estos anos son muchas las personas e instituciones que han participado en este

trabajo y a quienes quiero expresar mi gratitud por el apoyo y la confianza que me han

prestado de forma desinteresada.

En primer lugar quiero agradecer al Departamento de Matematicas de la Universidad

Autonoma por la calida acogida y el apoyo recibido. Gracias de corazon por hacerme un

pequeno sitio dentro de este departamento que ya considero mi casa.

Debo un especial reconocimiento a la Comunidad Autonoma de Madrid por la confianza

que mostraron en mı al concederme una beca FPI con la cual fue posible aventurame en

esta travesıa.

Un sincero agradecimiento a mi Director, Enrique Zuazua, ya que es el me ha dado la

oportunidad de introducirme en el mundo de la investigacion y ası poder realizar esta Tesis.

Me acuerdo del Profesor Xu Zhang, por todo el tiempo que me ha dado, por sus sug-

erencias de las que tanto provecho he sacado. Tambien agradezco al Profesor Arieh Iserles

durante una estancia breve en el Instituo de Issac Newton de Cambridge, a traves del

Programa HOP.

Gracias a Rafael Orive y Angel San Antolın por sus traduciones de Ingles a Espanol.

Todo esto nuca hubiera sido posible sin el amparo incondicional de mis padres y mis

abuelos y sin el amor de Xinyi. Este es tambien vuestro premio.

Contents

1. Introduccion y resultados principales 11.1. Controlabilidad de la ecuacion del calor discretizada en tiempo . . . . . . . 3

1.2. Controlabilidad de la ecuacion de ondas . . . . . . . . . . . . . . . . . . . . 61.3. Observabilidad del sistema lineal y conservativo . . . . . . . . . . . . . . . . 11

1. Introduction and main results 1

1.1. Controllability of the time-discrete heat equation . . . . . . . . . . . . . . . 31.2. Controllability of the time-discrete wave equation . . . . . . . . . . . . . . . 61.3. Observability of time-discrete conservative linear systems . . . . . . . . . . 11

2. Preliminaries 21

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2. Linear dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3. The heat equation and the L-R method . . . . . . . . . . . . . . . . . . . . 232.4. The wave equation: duality arguments . . . . . . . . . . . . . . . . . . . . . 262.5. Conservative linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3. The time-discrete heat equation 313.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2. Lack of controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3. Heuristics of the L-R method . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4. Partial null-controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5. Error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6. Approximate controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.7. Other discrete schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.7.1. The explicit Euler scheme . . . . . . . . . . . . . . . . . . . . . . . . 573.7.2. The θ-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.8. Time-discrete fractional order parabolic equations . . . . . . . . . . . . . . 62

4. The time-discrete wave equation 674.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3. Identities via multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.4. Hidden regularity and well-posedness . . . . . . . . . . . . . . . . . . . . . . 824.5. Lack of controllability/observability without filtering . . . . . . . . . . . . . 86

4.6. Uniform observability under filtering . . . . . . . . . . . . . . . . . . . . . . 88

i

4.6.1. Statement of the uniform observability result . . . . . . . . . . . . . 894.6.2. A technical result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.6.3. Proof of the uniform observability result . . . . . . . . . . . . . . . . 93

4.7. Optimality of the filtering parameter . . . . . . . . . . . . . . . . . . . . . . 944.7.1. Optimality of the order of the filtering parameter . . . . . . . . . . . 944.7.2. A heuristic explanation . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.8. Uniform controllability and convergence of the controls . . . . . . . . . . . . 99

5. Time-discrete conservative linear systems 1035.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2. The implicit mid-point scheme . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3. General time-discrete schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.3.1. General time-discrete schemes for first order systems . . . . . . . . . 1125.3.2. The Newmark method for second order in time systems . . . . . . . 116

5.4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.4.1. Application of Theorem 5.2.1 . . . . . . . . . . . . . . . . . . . . . . 1235.4.2. Application of Theorem 5.3.1 . . . . . . . . . . . . . . . . . . . . . . 1255.4.3. Application of Theorem 5.3.2 . . . . . . . . . . . . . . . . . . . . . . 127

5.5. Fully discrete schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.5.1. Main statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.5.2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.6. On the admissibility condition . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.6.1. The time-continuous setting . . . . . . . . . . . . . . . . . . . . . . . 1365.6.2. The time-discrete setting . . . . . . . . . . . . . . . . . . . . . . . . 140

6. Conclusions and open problems 143

ii

Resumen

En esta Tesis analizamos problemas del control de los esquemas discretizados en tiempode procesos de difusion y propagacion de ondas usando tecnicas del analisis numerico. Estu-diamos como estos esquemas aplicados a algunas ecuaciones en derivadas parciales afectana las propiedades de los modelos continuos, como la observabilidad, la controlabilidad, etc.

Para la ecuacion del calor estudiamos la controlabilidad a cero del esquema Euler im-plıcito discretizado en tiempo en un dominio acotado con un control interior. Usando elmetodo Lebeau-Robbiano de la iteracion en tiempo, demostramos que la proyeccion delcontrol sobre un espacio apropiadamente filtrado es controlable a cero por un control uni-formemente acotado con respecto al parametro de la malla del tiempo. Por eso, la bienconocida propiedad de controlabilidad de las ecuaciones en el caso continuo se puede pro-bar como un lımite, cuando la malla del tiempo se va a cero, de los controles de los casosdiscretizados en tiempo.

Tambien se han obtenido resultados similares para otros esquemas discretizados, comoel esquema Euler explicito, el θ-metodo, etc. Ademas, discutimos la controlabilidad a cerode la ecuacion parabolica de orden fraccional cuando discretizamos en tiempo por el metodode Euler implıcito, generando resultos similares al caso continuo.

Para la ecuacion de ondas, estudiamos la controlabilidad exacta en la frontera para ladiscretizacion en tiempo trapezoidal en un dominio acotado. Probamos que, usando tecnicasde multiplicadores, la proyeccion de la solucion en un apropiado espacio filtrado es exac-tamente controlable con un coste acotado uniformemente con respecto al paso de malla entiempo. De esta forma, la bien conocida propiedad de controlabilidad exacta de la ecuacionde ondas continua en tiempo puede ser reproducida como el lımite, cuando la malla deltiempo se va a cero, de las controlabilidades de las proyecciones de las soluciones. Por du-alidad, esos resultados son equivalentes a las estimaciones uniformes de la observabilidad(con respecto del paso de malla en tiempo) que se derivan dentro de las clases de solucionesdel problema discreto en tiempo en los cuales las componentes de alta frecuencia han sidofiltrados.

Por observabilidad puede ser establecido por medio de una version discretizada en tiempode tecnicas clasicas de multiplicadores. La optimalidad del orden del parametro de filtradotambien puede ser establecido, aunque un cuidadoso analisis de la velocidad de propagacionque las ondas discretizadas en tiempo, las cuales nos indican el valor exacto se puede mejorar.

El ultimo resultado en esta Tesis concierne a la observabilidad uniforme de varios meto-dos de discretizacion en tiempo de un sistema abstracto de evolucion zt = Az, donde Aes un operador skew-adjunto, y un operador de observacion B es dado. Mas precisamente,suponemos que el par (A,B) es exactamente observable en un nivel continuo, y obtenemosdesigualdades uniformes de observabilidad para varios metodos apropiados de discretizacionen tiempo dentro de una clase de datos iniciales convenientemente filtrados. Los metodos conlos que estamos trabajando en este capıtulo estan principalmente basados en la ası llama-da estimacion resolvente. Ademas, presentamos algunas aplicaciones de nuestros resultados

para metodos de discretizacion en tiempo para ecuaciones de ondas, ecuacion de KdV y unesquema completamente discretizado para la ecuacion de ondas. En particular, los resulta-dos previos sobre observabilidad uniforme en la frontera de la ecuacion de ondas dicretizadaen tiempo puede ser como una consecuencia directa de estos resultados abstractos.

Capıtulo 1

Introduccion y resultados

principales

El objetivo principal de esta tesis es analizar los problemas de control de esquemas de

aproximacion de procesos de difusion y propagacion de ondas usando tecnicas del analisis

numerico. Enfocaremos nuestro esfuerzo en los esquemas de discretizacion en tiempo. Mas

precisamente, estudiamos como estos esquemas aplicados a ecuaciones en derivadas parciales

afectan las propiedades de los modelos continuos, como la observabilidad, la controlabilidad,

etc.

Diversas aplicaciones industriales, como por ejemplo la construccion de aviones y de

plantas quımicas, han inspirado el origen y desarrollo de la Teorıa de Control. La Teorıa

de Control en sistemas finito-dimensionales fue introducida por Kalman ([41],[42]) y des-

de entonces, esta teorıa ha sido generalizada, primero a sistemas infinito-dimensionales y

despues a sistemas mucho mas complicados, por ejemplo, sistemas no lineales, sistemas de

parametros distributivos y sistemas estocasticos ([11], [15], [55], [57], [63], [87], [90]).

La teorıa de control se aplica a gran variedad de problemas. Uno de los mas naturales e

importantes es el problema de controlabilidad, que puede ser formulado como sigue. Consi-

deramos un sistema de evolucion (cualquiera descrito en terminos de ecuaciones diferenciales

ordinarias, EDOs o ecuaciones en derivadas pariciales, EDPs). Se nos permite actuar sobre

las trajectorias del sistema por algun metodo apropiado de control (la fuente o parte derecha

del sistema, las condiciones de contorno, etc). Entonces, para un intervalo de tiempo dado

t ∈ (0, T ), un dato inicial y otro final, tenemos que encontrar un control tal que se igualan

las dos soluciones, las del dato inicial en tiempo, t = 0, y las del dato final t = T . Existen

muchos textos en la literatura dedicados a estos temas. Nos referimos, por ejemplo, al libro

de Lee y Marcus [53], para una introduccion de esos modelos en el contexto de sistema

1

2 CAPITULO 1. INTRODUCCION Y RESULTADOS PRINCIPALES

finito-dimensionales. Ademas, nos referimos a un artıculo recopilatorio de Russell [87] y al

libro de Lions [55] (ademas, de su artıculo [57]) para una introduccion a la controlabilidad

de las EDPs. Tambien, para conocer el estado del arte consideramos los siguientes artıculos:

[66], [107], [109] y [112].

En 1988, Lions en el libro [57] introdujo el llamado Metodo de Unicidad de Hilbert

(HUM). Hablando de manera informal, esta basado en el principio de que siempre un sis-

tema es controlable y se puede construir el control minimizando un apropiado funcional

cuadratico definido sobre una clase de soluciones del sistema adjunto. Determinadas vari-

antes de este funcional permiten construir controles de norma mınima en L2, controles

bang-bang, controles aproximados, etc. La principal dificultad cuando se minimizan estos

funcionales es mostrar que son coercivos. Esto resulta ser equivalente a la llamada propiedad

de observabilidad de la ecuacion adjunta, la cual indica que se puede determinar univoca-

mente los estados adjuntos en cualquier lugar en terminos de medidas parciales, por ejemplo,

evaluar la energia del sistema total via medidas de frontera.

La controlabilidad y la observabilidad de los sistemas continuos son bien entendidas

para modelos lineales escalares y, por lo tanto, para construir esquemas de aproximacion

numerica eficientes, se necesita tratar con temas similares para apropiados esquemas de

discretizacion. Pero es bien conocido que la discretizacion puede romper las propiedades

de controlabilidad o observabilidad del sistema continuo. Por ejemplo, la interaccion de

ondas con una malla numerica produce fenomenos de dispersion y falsas oscilaciones de

altas frecuencias ([94], [97]). En particular, debido a esta interaccion no fısica de ondas

con el medio discreto, la velocidad de propagacion de las soluciones numericas de ondas,

la llamada velocidad de grupo, debe converger a cero cuando la longitud de onda de las

soluciones tiene un orden como el de la anchura de la malla y la hacemos tender a cero.

Como consecuencia de este fenomeno, el tiempo, que se necesita para observar (o controlar)

uniformemente (con respecto del tamano de la malla) las ondas numericas desde la frontera

o desde un subconjunto del dominio en el cual se propagan, debe tender a infinito cuando

la malla se va haciendo fina. Por lo tanto, las propiedades de observacion y de control del

modelo discreto deben eventualmente desaparacer.

El analisis de la propiedades de controlabilidad y observabilidad de los esquemas numeri-

cos de aproximacion han sido el objeto de un intenso estudio aunque la mayorıa de los

resultados que aparecen en la literatura afectan al caso de la semi-discretizacion de espacios

([5], [25], [32], [60], [73], [106] y [110]). El principal trabajo desarrollado en esta tesis es sobre

CAPITULO 1. INTRODUCCION Y RESULTADOS PRINCIPALES 3

esquemas de discretizacion de tiempo y sus propiedades de controlabilidad y observabilidad.

Hablando de una manera informal, nuestra atencion se deposita en encontrar situaciones

apropiadas tales que, bajo ciertas consideraciones, las propiedades de discretizacion en tiem-

po llegan a ser uniformes, es decir, el lımite se adapta al caso continuo tanto como en las

propiedades de controlabilidad como en las de observabilidad.

En esta tesis resolvemos los siguientes tres problemas:

1. Problema de control en el interior del dominio de la ecuacion del calor discretizada en

tiempo.

2. Controlabilidad y observabilidad en la frontera de la ecuacion de ondas discretizada

en tiempo.

3. Observabilidad de sistemas lineales conservativos discretizados en tiempo.

Sigamos ahora describiendo mas precisamente los problemas que estamos estudiando en

esta tesis, los resultados principales obtenidos y el enfoque que hacemos de estos resultados.

1.1. Controlabilidad de la ecuacion del calor discretizada en

tiempo

En el capıtulo 3 vamos a analizar unas propiedades de la controlabilidad interior de la

ecuacion del calor discretizada en tiempo en un dominio acotado.

Sea Ω un dominio abierto y acotado en lRd (d ∈ lN∗) con una frotena ∂Ω = Γ en C∞.

Sea ω un subconjunto no vacıo del Ω. Consideramos la ecuacion del calor con un control

interior:

yt − ∆y = u1ω, (t, x) ∈ (0, T ) × Ω

y = 0, (t, x) ∈ (0, T ) × Γ

y(0, x) = y0(x), x ∈ Ω.

(1.1)

Es bien conocido que el sistema (1.1) es controlable a cero para todo T > 0 y todos los

subconjuntos ω ⊂ Ω abiertos y no vacıos (vease, por ejemplo, [23] y [49]), es decir, para

todo y0 ∈ L2(Ω) tenemos un control u ∈ L2((0, T )×ω), tal que la solucion correspondiente

de (1.1) satisface

y(T, x) = 0, ∀ x ∈ Ω.


Ademas, tenemos la siguiente estimacion para el control a cero u que minimiza la norma

del L2 del sistema (1.1):

‖u‖L2((0,T )×ω) ≤ C∥∥y0∥∥

L2(Ω),

donde C es una constante positivo depende solo por T,Ω y ω. De ahora y en adelante,

usamos C para denotar una constante positiva generica (independiente de los parametros

de la discretizacion en espacio o tiempo).

La propiedad de controlabilidad aproximada se puede formular de modo parecido, es

decir, para todo y0, y1 ∈ L2(Ω) y ε > 0, buscamos un control u ∈ L2((0, T ) × ω) tal que la

solucion de (1.1) satisface∥∥y(T ) − y1

∥∥L2(Ω)

≤ ε.

Dada K ∈ lN∗, consideramos t = T/K e introducimos el mallado

t0 = 0 < t1 < · · · < tK = T

con tk = kt y k = 0, 1, · · · ,K.

La version discreta en tiempo del sistema (1.1) se lee:

yk+1 − yk

t − ∆yk+1 = uk1ω, x ∈ Ω, k = 0, 1, · · · ,K − 1,

yk = 0, x ∈ Γ, k = 1, · · · ,K,y0 ∈ L2(Ω) dada.

(1.2)

El sistema (1.2) es una discretizacion implıcita en tiempo de la ecuacion del calor con

control en el subconjunto ω ⊂ Ω. Aquı, yk0≤k≤K representan los estados y uk0≤k≤K−1

los controles.

Introduzcamos la propiedad de controlabilidad nula para este sistema discreto en tiempo:

El sistema (1.2) se dice que es controlable a cero en tiempo T (para cualquier t > 0 dado)

si para cualquier y0 ∈ L2(Ω) existe un control uk ∈ L2(ω)0≤k≤K−1 con K = T/t, que a

partir de ahora llamamos control discreto a cero, tal que la solucion ykk=0,··· ,K de (3.1)

satisface

yK(x) = 0, ∀ x ∈ Ω.

Tambien podemos definir de forma similar la propiedad de controlabilidad aproximada

de un sistema discreto en tiempo como: El sistema (1.2) se dice que es controlable apro-

ximadamente en tiempo T si para cualquier y0 ∈ L2(Ω), cualquier estado final yT ∈ L2(Ω)

y ε > 0, existe un control uk ∈ L2(ω)0≤k≤K−1 tal que la solucion ykk=0,··· ,K de (1.2)

satisface que∥∥yK − yT

∥∥L2(Ω)

< ε.

El primer resultado de control para el sistema (1.2) es tal que, sea quien sea Ω ⊂ lRd,

estas propiedades de controlabilidad del sistema (1.2) no se cumplen.


Theorem 1.1.1. Sea ω un subconjunto no vacio de Ω. Para cualquier t > 0 dado, el

sistema (1.2) no es ni controlable a cero ni aproximadamente controlable.

En vista de la falta de controlabilidad para el sistema (1.2) es natural precisar de algunos

requirimientos para obtenerla. En particular, vamos a estudiar la controlabilidad de las

proyecciones de las soluciones sobre una apropiada clase de componentes de Fourier de

bajas frecuencias. Por esta razon, introducimos los siguientes espacios filtrados basados en

la descomposicion de Fourier.

Sea Φj ∈ H10 (Ω) una base ortonormal de L2(Ω) formada por las autofunciones del

Laplaciano con condiciones Dirichlet:

−∆Φj = µ2jΦj, x ∈ Ω

Φj = 0, x ∈ Γ.(1.3)

Para s > 0, definimos

C1,s = f(x) | f(x) =∑

µ2j<s

ajΦj(x), aj ∈ lC ⊂ H10 (Ω), (1.4)

C0,s = g(x) | g(x) =∑

µ2j<s

bjΦj(x), bj ∈ lC ⊂ L2(Ω), (1.5)

y

C−1,s = z(x) | z(x) =∑

µ2j<s

cjΦj(x), cj ∈ lC ⊂ H−1(Ω), (1.6)

subespacios de H10 (Ω), L2(Ω) y H−1(Ω), respectivamente, con las topologıas. Es claro que

∞⋃

k=1

C1,k es denso en H10 (Ω), y lo mismo se puede decir para

∞⋃

k=1

C0,k en L2(Ω) y∞⋃

k=1

C−1,k

en H−1(Ω). Denotamos por π1,s, π0,s y π−1,s los operadores proyeccion de H10 (Ω), L2(Ω) y

H−1(Ω) a C1,s, C0,s y C−1,s, respectivamente. Los espacios C±1,s y los proyecciones π±1,s no

se utilizaran en esta seccion pero seran necesarios mas adelante.

Un nuevo resultado para el sistema (1.2) muestra que la proyeccion de la solucion de

(1.2) en C0,s es uniformemente controlable con una apropiada eleccion de los valores del

parametro de filtrado s, en particular, como funcion del parametro de dicretizacion t:

Theorem 1.1.2. Dado T > 0 fijo y r ∈ (0, 2), existe una constante positiva Λ = Λ(r, T,Ω, ω)

tal que para todo y0 ∈ L2(Ω), existe un control uk ∈ L2(ω)k=0,··· ,K−1, tal que

(1) La solucion del sistema (1.2) satisface

π0,Λ(t)−ryK(x) = 0, ∀ x ∈ Ω; (1.7)


(2) Existe una constante C = C(r, T,Ω, ω) > 0, independiente de t, tal que se tiene

tK−1∑

k=0

∫

ω|uk|2dx ≤ C

∫

Ω|y0|2dx (1.8)

para cualquier t > 0 y y0 ∈ L2(Ω).

La principal tecnica usada en la demostracion del Teorema 1.1.2 es una analoga al

metodo de iteracion en tiempo de Lebeau-Robbiano en el nivel de tiempo-discreto, veanse

los detalles en la Seccion 3.4.

Ademas, tenemos la siguiente propiedad de convergencia y estimcion de los errores para

los controles a cero discretos:

Theorem 1.1.3. Para el control discreto a cero uk0≤k≤K−1 dado en la demostracion del

Teorema 1.1.2, se tiene

UK(·, x) =

K−1∑

k=0

uk(x)1[tk ,tk+1)(·) −→ u(·, x)

fuertemente en L2((0, T ) × ω) cuando t → 0, donde u es un control a cero del sistema

(1.1). Por otra parte, existe una constante C > 0, independiente de t y y0, tal que UK y

u satisfacen∥∥UK − u

∥∥L2((0,T )×ω)

≤ C√t∥∥y0∥∥

L2(Ω). (1.9)

Tambien, en la ultima parte del Capıtulo 3, discutimos otros esquemas de discretizacion

en tiempo y sus correspondientes propiedades de controlabilidad para la ecuacion del calor

(1.1), como es el caso, del metodo implıcito de Euler, del θ-metodo, etc. De manera si-

milar, analizamos la discretizacion implıcita en tiempo de orden fraccional para el sistema

parabolico y obtenemos resultados similares.

1.2. Controlabilidad de la ecuacion de ondas discretizada en

tiempo

En el Capıtulo 4, estudiamos las propiedades de controlabilidad exacta en la frontera

para la ecuacion de ondas con discretizacion en tiempo trapezoidal en un dominio acotado.

Sea Ω definido como antes, Γ0 un conjunto no vacio de Γ, y T > 0 un tiempo final dado.


Consideramos la siguiente ecuacion de ondas (continua en tiempo) con una funcion

control actuando en el subconjunto Γ0 de la frontera:

ytt − ∆y = 0 en (0, T ) × Ω

y = u1Γ0 en (0, T ) × Γ

y(0) = y0, yt(0) = y1 en Ω.

(1.10)

En (1.10), (y(t, ·), yt(t, ·)) es el estado y u(t, ·) es el control. Los espacios de estado y con-

trol del sistema (1.10) que eligimos son L2(Ω)×H−1(Ω) y L2((0, T )×Γ0), respectivamente.

La propiedad de controlabilidad exacta en la frontera de (1.10) se define como sigue:

Para cualquier (y0, y1) ∈ L2(Ω) ×H−1(Ω), existe un control u ∈ L2((0, T ) × Γ0) tal que la

solucion y ∈ C([0, T ];L2(Ω)) ∩ C1([0, T ];H−1(Ω)) de (1.10), definida por el metodo clasico

de transposicion ([57]), satisface:

y(T ) = yt(T ) = 0 en Ω. (1.11)

A diferencia del caso de las ecuaciones del calor, la propiedad de controlabilidad para la

ecuacion de ondas (1.10) se tiene solo bajo unas ciertas restricciones geometricas en el con-

junto Γ0 de la frontera donde actua el control y necesitando de un tiempo de controlabilidad

T suficientemente grande.

Por clasicos argumentos de dualidad ([57]), la anterior propiedad de controlabilidad es

equivalente a una estimacion de observabilidad en la frontera de la siguiente ecuacion de

ondas sin ningun termino de control:

ϕtt − ∆ϕ = 0, en (0, T ) × Ω

ϕ = 0 en (0, T ) × Γ

ϕ(T ) = ϕ0, ϕt(T ) = ϕ1, en Ω.

(1.12)

En particular, la deigualdad de observabilidad es la siguiente:

E(0) ≤ C

∫ T

0

∫

Γ0

∣∣∣∂ϕ

∂ν

∣∣∣2dΓ0dt, ∀ (ϕ0, ϕ1) ∈ H1

0 (Ω) × L2(Ω). (1.13)

Por otro lado, E(0) representa la energıa E(t) de (1.12) en t = 0, con

E(t) =1

2

∫

Ω

[|ϕt(t, x)|2 + |∇ϕ(t, x)|2

]dx. (1.14)

La desigualdad (1.13) nos dice que la energıa total para la solucion de (1.12) se puede

observar en terminos de la energıa concentrada en Γ0 en el intervalo de tiempo (0, T ). Es

bien conocido que hay tipicamente dos clases de condiciones en (T,Ω,Γ0) que garantizan


(1.13). La primera esta dada por la clasica condicion de multiplicadores. Mas precisamente,

fijado un x0 ∈ lRd, tenemos

Γ0=x ∈ Γ

∣∣ (x− x0) · ν(x) > 0, R

= max

x∈Ω|x− x0|, (1.15)

donde ν(x) es el vector normal unitario hacia afuera de Ω en x ∈ Γ. Entonces (1.13) se tiene

para Γ0 como en (1.15) tomando T > 2R. Esta es la tıpica situacion que uno se encuentra

aplicando el metodo de los multiplicadores ([57]), y las desigualdades de Carleman (vease

[102]) para deducir (1.13). El segundo es cuando (T,Ω,Γ0) satisface la Condicion Geometrica

de Control (GCC, para abreviar) introducida en [2], la cual afirma que todos los rayos de

la optica geometrica en Ω intersectan la parte de la fontera Γ0 en un tiempo uniforme T .

En este caso, (1.13) se establece por medio de de herramietas de analisis micro-local ([2]).

Esta condicion es optima.

Ahora introducimos la siguiente semi-discretizacion trapezoidal en tiempo de (1.10):

yk+1 + yk−1 − 2yk

(t)2 − ∆(yk+1 + yk−1

2

)= 0,

en Ω, k = 1, · · · ,K − 1

yk = uk1Γ0 , en Γ, k = 0, · · · ,Ky0 = y0, y1 = y0 + ty1.

(1.16)

Aquı (y0, y1) ∈ L2(Ω) × H−1(Ω) son los datos dados en el sistema (1.10) que permiten

determinar los datos iniciales para el sistema discreto en tiempo. Probamos que el sistema

discreto (1.16) esta bien puesto por medio del metodo de transposicion.

El problema de controlabilidad para el sistema discreto (1.16) se puede formular como

sigue: Para cualquier par (y0, y1) ∈ L2(Ω) ×H−1(Ω), encontrar un control

uk ∈ L2(Γ0)k=1,··· ,K−1

tal que la solucion ykk=0,··· ,K de (1.16) satisface:

yK−1 = yK = 0 en Ω. (1.17)

Notemos que (1.17) es equivalente a la condicion yK−1 = (yK − yK−1)/t = 0 que es la

discretizacion natural de (1.11).

Como en el contexto de la ecuacion de ondas continua anterior, tambien vamos a con-


siderar el sistema no controlado

ϕk+1 + ϕk−1 − 2ϕk

(t)2 − ∆

(ϕk+1 + ϕk−1

2

)= 0,

en Ω, k = 1, · · · ,K − 1

ϕk = 0, en Γ, k = 0, · · · ,KϕK = ϕt

0 + tϕt1 , ϕK−1 = ϕt

0 , en Ω,

(1.18)

donde (ϕt0 , ϕt

1 ) ∈ (H10 (Ω))2. En particular, para garantizar la convergencia de las solu-

ciones de (1.18) hacia las del problema (1.12) suponemos que

ϕt

0 → ϕ0 fuertemente en H10 (Ω),

ϕt1 → ϕ1 fuertemente en L2(Ω).

cuando K → ∞, o t→ 0, (1.19)

con tϕt1 siendo acotado en H1

0 (Ω). Obviamente este hecho es siempre posible por la

compacidad de H10 (Ω) en L2(Ω).

La energıa del sistema (1.18) esta dada por

Ekt

=

1

2

∫

Ω

(∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

+|∇ϕk+1|2 + |∇ϕk|2

2

)dx, k = 0, · · · ,K − 1, (1.20)

la cual es la definicion analoga a la energıa E(t) del problema continuo definida en (1.14).

Multiplicando la primera ecuacion del sistema (1.18) por (ϕk+1 −ϕk−1)/2 e integrando por

partes en Ω vemos facilmente la siguiente propiedad de conservacion de la energıa:

Ekt = E0

t, k = 0, · · · ,K − 1. (1.21)

Consecuentemente el esquema bajo consideracion es estable y su convergencia (en el sen-

tido clasico del analisis numerico) se obtiene para una apropiada eleccion de los espa-

cios funcionales de acuerdo con la definicion de energıa elegida, en particular, el espacio

H10 (Ω) × L2(Ω) dada la condicion (1.19)) considerada.

Por medio de clasicos argumentos de dualidad, es facil de ver que la anterior propiedad

de controlabilidad (1.17) es equivalente a la siguiente propiedad de observabilidad en la

frontera para las soluciones ϕkk=0,··· ,K de (1.18):

E0t ≤ Ct

K−1∑

k=1

∫

Γ0

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ0, ∀ (ϕt0 , ϕt

1 ) ∈ (H10 (Ω))2. (1.22)

Igual que en Teorema 1.1.1, tenemos el siguiente resultado negativo para los sistemas

(1.16)-(1.18):


Theorem 1.2.1. Dado cualquier t > 0 y cualquier subconjunto abierto no vacio Γ∗ de

Γ, el sistema (1.18) no es observable, y por lo tanto, el sistema (1.16) no es controlable a

cero.

Nuestro resultado de observabilidad uniforme para el sistema (1.18) requiere de un

adecuado filtrado de las componenetes de altas frecuencias y lo presentamos de la siguiente

manera:

Theorem 1.2.2. Sea T > 2R. Entonces existen dos constantes positivas h0 > 0 y δ > 0,

dependiendo solo de la dimension, T y R, tal que para todo par (ϕt0 , ϕt

1 ) ∈ C1,δ(t)−2 ×C0,δ(t)−2 , la correspondiente solucion ϕkk=0,··· ,K de (1.18) satisface

E0t ≤ Ct

K−1∑

k=1

∫

Γ0

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ0, (1.23)

para todo t ∈ (0, h0].

La principal herramineta para probar el Teorema 1.2.2 es un metodo de multiplicadores

discreto en tiempo. Vease para mas detalles la Seccion 4.6.

Como consecuencia del Teorema 1.2.2, tenemos los siguientes resultados de controlabi-

lidad uniforme para el sistema (1.16) y de convergencia para los controles:

Theorem 1.2.3. Sean T , h0 y δ dados como en Teorema 1.2.2, y K > 1un entero impar.

Entonces para cualquier t ∈ (0, h0] y par de datos (y0, y1) ∈ L2(Ω) ×H−1(Ω), existe un

control uk ∈ L2(Γ0)k=1,··· ,K−1 tal que la solucion de (1.16) satisface

π0,δ(t)−2yK−1 = π−1,δ(t)−2

(yK − yK−1

t)

= 0 en Ω; (1.24)

Existe una constante C > 0, independiente de t, y0 y y1, tal que

tK−1∑

k=1

∥∥∥uk∥∥∥

2

L2(Γ0)≤ C

∥∥∥∥(y0,

y1 − y0

t

)∥∥∥∥2

L2(Ω)×H−1(Ω)

; (1.25)

Cuando t→ 0,

Ut=

K−1∑

k=1

uk(x)1[kt,(k+1)t)(t) −→ u fuertemente en L2((0, T ) × Γ0), (1.26)

donde u es un control del sistema (1.10), cumpliendo (1.11);


Cuando t→ 0,

yt= y010(t) +

1

t

K−1∑

k=0

[(t− kt)yk+1 −

(t− (k + 1)t

)yk]1(kt,(k+1)t](t)

−→ y fuertemente en C([0, T ];L2(Ω)) ∩H1([0, T ];H−1(Ω)),

(1.27)

donde y es la solucion de (1.10) con el control u obtenido en el lımite anterior.

Ademas, mostramos el siguiente resultado de optimabilidad, el cual nos dice que el orden

(t)−2 del parametro de filtrado en el Teorema 1.2.2 es optima.

Theorem 1.2.4. Supongamos que Γ∗ es cualquier subconjunto abierto no vacio de Γ. En-

tonces, dado cualquier a > 2, se sigue que

lımt→0

sup(ϕt

0 ,ϕt1 )∈C1,(t)−a×C0,(t)−a

E0t

tK−1∑

k=1

∫

Γ∗

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ∗

= ∞. (1.28)

Notar que, el Capıtulo 4 se ocupa de controles en la frontera, en vez de controles en el

interior los cuales se analizan en el capıtulo previo. De hecho, en este capıtulo se obtiene

facilmente, como consecuencia de los resultados de control en frontera, resultados para el

caso donde el control actua en un entorno de un subconjunto de la frontera de la forma de

Γ0.

1.3. Observabilidad del sistema lineal y conservativo discretiza-

do en tiempo

En el capıtulo 5 vamos a analizar las propiedades de observabilidad de varios sistemas

lineales y conservativos discretizados en tiempo.

Sea X un espacio de Hilbert dotado de la norma ‖·‖X y A : D(A) → X un operador

anti-adjunto de resolvente compacta. Consideramos la siguiente ecuacion abstracta:

zt(t) = Az(t), z(0) = z0. (1.29)

En (1.29), z0 ∈ X es el estado inicial, z = z(t) es el estado. Estos sistemas se u-

san a menudo como modelos de sistemas vibrantes (por ejemplo, la ecuacion de ondas),

de fenomenos electromagneticos (la ecuacion de Maxwell) o de la mecanica cuantica (la

ecuacion de Schrodinger), etc.


Supongamos que Y es otro espacio de Hilbert equipado con la norma ‖·‖Y . Denotamos

L(X,Y ) como el espacio de los operadores lineales y acotados de X a Y , dotado de la norma

clasica para este espacio. Sea B ∈ L(D(A), Y ) un operador de observacion y definimos la

funcion salida

y(t) = Bz(t). (1.30)

Para dar un sentido a (1.30), consideramos la hipotesis de que B es un operador de

observacion en el siguiente sentido (vease [98]): B se dice que es admisible para cualquier

T > 0, si existe una constante KT ≥ 0 tal que

∫ T

0‖y(t)‖2

Y dt ≤ KT ‖z0‖2X ∀ z0 ∈ D(A). (1.31)

Observese si B es acotado, es decir se puede extender de manera que B ∈ L(X,Y ),

entonces claramente B es un operador de observacion admisible.

La propiedad observabilida del sistema (1.29)-(1.30) se puede formular como sigue: El

sistema (1.29)-(1.30) se dice que es observable en tiempo T si existe kT ≥ 0 tal que

kT ‖z0‖2X ≤

∫ T

0‖y(t)‖2

Y dt ∀ z0 ∈ D(A). (1.32)

Ademas, (1.29)-(1.30) se dice que es observable si es observable para algun tiempo T > 0.

Notese que desde que observabilidad y contrabilidad son nociones duales, cada declaracion

que afecta a las propiedades de observabilidad tiene su homologo en el tema de contrabil-

idad. Por eso, en este capıtulo, solo nos centraremos en las propiedades de observabilidad

del sistema (1.29)-(1.30).

Durante el Capıtulo 5, supondremos que el sistema continuo (1.29)-(1.30) es observable.

Por consiguiente, segun la prueba de Hautus, la pareja (A,B) satisface una condicion inde-

pendiente de T ([3], [70]). Mas precisamente, esta demostrado en [70] que bajo la propiedad

de observabilidad tenemos la siguiente estimacion de la resolvente:

Existe dos constantes M,m > 0 tal que

M2 ‖(iωI −A)z‖2 +m2 ‖Bz‖2Y ≥ ‖z‖2 , ∀ ω ∈ lR, z ∈ D(A).

(1.33)

Notese que es muy natural obtener que sistemas continuos del tipo de (1.29)–(1.30)

son observables, como las ecuaciones de ondas, de placas, de Schorodinger, de elasticidad,

etc., usando difrentes metodos incluyendo el analisis de microlocal, multiplicadores, y series

de Fourier, etc. Nuestro objetivo es desarrollar una teorıa permitiendo que nos permite

obtener los resultados de controlabilidad para los sistemas discretizados en tiempo como

una consecuencia directa de los sistemas continuos en tiempo correspondientes.


Ası, consideramos la semi-discretizacion en tiempo del sistema (1.29)–(1.30). Sustituimos

la ecuacion de evolucion continua (1.29) y la desigualidad de observabilidad correspondiente

(1.32) por las del caso discretizado en tiempo.

Para ello introducimos el esquema de punto medio implıcito para discretizar el sistema

(1.29) en tiempo:

zk+1 − zk

t = A(zk+1 + zk

2

), en X, k ∈ Z

z0 dado.

(1.34)

La funcion salida de (1.34) viene dada por

yk = Bzk, k ∈ Z. (1.35)

Notese que (1.34)–(1.35) es una version discreta del sistema (1.29)–(1.30).

Como A es un operador imaginario puro, obviamente tenemos que la norma de zk enX se

conserva con respecto al parametro discretizado en tiempo k ∈ Z, es decir,∥∥zk∥∥

X=∥∥z0∥∥

X.

Por eso, el esquema que consideramos es estable y su convergencia (en el sentido clasico del

analisis numerico) se puede garantizar en un entorno funcional apropiado.

El problema de la observabilidad uniforme del sistema (1.34) se puede formular como

sigue: Buscar una constante kT , independiente de t, tal que las soluciones zk del sistema

(1.34) satisfacen:

kT

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥Bzk∥∥∥

2

Y, (1.36)

para todos los datos iniciales z0 en una clase apropiada.

Claramente, (1.36) es una version discreta de (1.32). Nos interesa entender, con que

condiciones, la desigualdad (1.36) se verifica uniformemente en t. Esperamos poder hacerlo

de manera que cuando pasamos al l’ımite t a 0 recuperamos la propiedad de observabilidad

del modelo continuo.

Esto tambien se puede hacer por medio de tecnicas de filtrado espectral. Mas pre-

cisamente, como A es anti-adjunto con resolvente compacta, su espectro viene dado por

σ(A) = iµj : j ∈ Λ donde Λ = Z∗ o N∗ y (µj)j∈Λ es una sucesion de numeros reales. Sea

(Ψj)j∈Λ la base ortonormal asociada a las autofunciones deA con sus respectivos autovalores

(iµj)j∈Λ, es decir:

AΨj = iµjΨj . (1.37)

Por eso, como en (1.4)–(1.6), para todo s > 0, definimos

Cs = span Ψj : tal que su correspondiente iµj satisface |µj | ≤ s. (1.38)


Hay que notar que el espacio de Hilbert D(A) esta dotado con la norma del grafo de A

siguiente:

‖z‖21 = ‖z‖2

X + ‖Az‖2X .

El hecho de que B ∈ L(D(A), Y ), implica que

‖Bz‖Y ≤ CBδ

t ‖z‖X , z ∈ Cδ/t, (1.39)

donde CB es una constante positiva independiente de t.Demostraremos que la desigualdad (1.36) se verifica uniformemente (con respecto de

t > 0) en la clase Cδ/t para todo δ > 0 y todo Tδ suficiente grande, que depende del

parametro filtrado δ. Eso se obtiene como una consecuencia del siguiente Teorema:

Theorem 1.3.1. Dado δ > 0. Supongamos que tenemos una sucesion en el espacio vectoral

Xδ,t ⊂ X y una sucesion de operadores no acotados (At, Bt) tal que

(H1) Para todo t > 0, el operador At es anti-adjunto sobre Xδ,t, y el espacio vectorial

Xδ,t es globalmente invariante por At. Ademas,

‖Atz‖X ≤ δ

t ‖z‖X , ∀z ∈ Xδ,t, ∀t > 0. (1.40)

(H2) Existe una constante positiva CB tales que

‖Btz‖Y ≤ CB ‖Atz‖X , ∀z ∈ Xδ,t, ∀t > 0. (1.41)

(H3) Existe dos constantes M y m tal que

M2 ‖(At − iωI)z‖2X +m2 ‖Btz‖2

Y ≥ ‖z‖2X ,

∀z ∈ Xδ,t ∪D(At),∀ω ∈ lR, ∀t > 0.(1.42)

Entonces existe un tiempo Tδ, tal que para cualquier T > Tδ, existe una constante kT,δ, tal

que para t suficiente pequeno, la solucion de

zk+1 − zk

t = At

(zk+1 + zk

2

), en Xδ,t, k ∈ Z, . (1.43)

con datos iniciales z0 ∈ Xδ,t, satisface

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥Btzk∥∥∥

2

Y, ∀ z0 ∈ Xδ,t. (1.44)


Ademas, comparando con el tiempo optimo de observabilidad T0 = πM en el caso continuo,

Tδ se puede obtener de la manera

Tδ = π[(

1 +δ2

4

)2M2 +m2C2

B

δ4

16

]1/2, (1.45)

donde CB es como en (1.39).

La prueba del Teorema 1.3.1 aparece en el Capıtulo 5 y los conceptos clave son el uso

de la transformada de tiempo discreto y la estimacion de la resolvente.

De hecho, dados At = A, Bt = B y Xδ/t = Cδ/t, obviamente el Teorema 1.3.1

proporciona un resultado de observabilidad para el sistema (1.34)–(1.35) en la clase Cδ/t,

como el que sigue:

Theorem 1.3.2. Supongamos que (A,B) satisface (1.33) y B ∈ L(D(A), Y ). Entonces,

para cualquier δ > 0, existen Tδ y t0 > 0 tales que para todo T > Tδ y t ∈ (0,t0),existe una constante positiva kT,δ, independiente de t, tal que la solucion zk de (1.34)

satisface

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥Bzk∥∥∥

2

Y, ∀ z0 ∈ Cδ/t. (1.46)

Ademas, Tδ se puede escojer de la manera

Tδ = π[M2(1 +

δ2

4

)2+m2C2

B

δ4

16

]1/2, (1.47)


El Teorema 1.3.1 tambien sirve para estudiar la observabilidad de la discretizacion en

tiempo de los problemas (1.29)-(1.30) mas generales. Por ejemplo, consideramos un esquema

discreto en tiempo de (1.29)-(1.30) como

zk+1 = Ttzk, yk = Bzk, (1.48)

donde Tt es un operador lineal cuyas autofunciones son las de A. Tambien suponemos que

este sistema es conservativo en el sentido de que existen numeros reales λj,t tales que

TtΨj = exp(iλj,tt)Ψj . (1.49)

Ademas, suponemos que existe una relacion explıcita entre λj,t y µj de la forma siguiente:

λj,t =1

t h(µjt), (1.50)


donde h : [−δ, δ] 7→ [−π, π] es una funcion suave estrictamente creciente, es decir,

|h(η)| ≤ π, ınfh′(η), |η| ≤ δ > 0. (1.51)

Hablando de manera aproximada, el primer termino de (1.51) indica que no se pueden

evaluar las frecuencias superiores a π/t con la malla de tamano t. Ademas, el segundo

termino es una condicion de no degeneracion para la velocidad del grupo ([94]) de las

soluciones de (1.48), lo que es necesario para garantizar la progagacion de las soluciones y

verificar la observabilidad.

Tambien suponemos que

h(η)

η−→ 1 cuando η → 0, (1.52)

lo cual es natural por la consistencia del esquema discretizado en tiempo con el caso continuo

(2.18).

Ahora mostramos el segundo resultado del Capıtulo 5, en el que demostramos que para

cualquier δ > 0, la desigualdad (1.36) se verifica uniformemente de t para todas las

soluciones de (1.48) cuando los datos iniciales se toman en la clase Cδ/t:

Theorem 1.3.3. Con las hipotesis (1.49), (1.50) y (1.51), para todo δ > 0, existe un

tiempo Tδ tal que para todo T > Tδ, existe una constante kT,δ > 0 tal que para todo tsuficientemente pequeno, cualquier solucion de (1.48) con dato inicial z0 ∈ Cδ/t satisface

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥∥B(zk + zk+1

2

)∥∥∥∥2

Y

. (1.53)

Ademas, tenemos la siguiente estimacion sobre Tδ:

T 2δ < 2π2

[2 tan2

(h(δ)2

)+ C2

Bδ2m2

(1 + tan2

(h(δ)2

))

+ 2M2(

ınf|ω|≤δ

∣∣∣(1 + tan2

(h(δ)2

))h′(ω)

∣∣∣)−2(

1 + tan4(h(δ)

2

))]. (1.54)


La demostracion del Teorema 1.3.3 es similar a la del Teorema 1.3.1, vease con mas

detalles en la Seccion 5.3.1.

El ultimo resultado del Capıtulo 5 trata sobre un sistema con segunda derivada en

tiempo. Sea H un espacio de Hilbert dotado con la norma ‖·‖H , y A0 : D(A0) → H un

operador autoadjunto con resolvente compacto. Consideramos el problema de dato inicial

utt +A0u = 0,

u(0) = u0, ut(0) = v0,(1.55)


que se puede interpretar como un modelo generico para vibraciones libres de estructuras de

cuerdas, rayos, membranas, placas o cuerpos tridimensionales elasticos, etc.

Por supuesto, estos sistemas se pueden escribir como sistemas de primer orden como

(1.29). Sin embargo, existen unos esquemas discretos en tiempo como el metodo de New-

mark, que no se puede poner de la forma (1.49). Por eso es natural analizar el esquema

discreto en tiempo para (1.55). Obtenemos una extension natural del Teorema 1.3.1 para

esta clase de sistemas de segundo orden (en tiempo).

La energıa de (1.55) es

E(t) = ‖ut(t)‖2H +

∥∥∥A1/20 u(t)

∥∥∥2

H, (1.56)

la cual es constante con respecto del tiempo.

Consideramos la funcion salida

y(t) = B1u(t) +B2ut(t), (1.57)

donde B1 y B2 son dos operadores de observalidad satisfaciendo que B1 ∈ L(D(A0), Y ) y

B2 ∈ L(D(A1/20 ), Y ). Es decir, suponemos que existe dos constantes CB,1 y CB,2, tal que

‖B1u‖Y ≤ CB,1 ‖A0u‖H , ‖B2v‖Y ≤ CB,2

∥∥∥A1/20 v

∥∥∥ . (1.58)

Vamos a suponer que B1 = 0 o B2 = 0. De hecho, esto es condicion necesaria en el estudio

del problema discretizado en tiempo debido a un problema tecnico que no hemos podido

resolver hasta ahora.

El sistema (1.55)–(1.57) se puede escribir de la forma (1.29)–(1.30). De hecho, dado

z1(t) = ut + iA1/20 u, z2(t) = ut − iA

1/20 u, (1.59)

la ecuacion (1.55) es igual que

zt = Az, z =

(z1z2

), A =

(iA

1/20 0

0 −iA1/20

). (1.60)

En (1.60) el espacio de energıa es X = H2 con el dominio D(A) = D(A1/20 )2. Ademas, la

energıa dada en (1.56) es la mitad de la norma de z en X.

Notese que el espectro de A es explıcito y esta dado por el espectro de A0. Realmente,

(µ2j )j∈N∗ (µj > 0) son los autovalores de A0, es decir,

A0φj = µ2jφj , j ∈ N

∗,


con las autofunciones φj asociadas correspondientes, ya que los autovalores de A son ±iµj

y sus autofunciones asociadas son respectivamente

Ψj =

(φj

0

), Ψ−j =

(0

φj

), j ∈ lN∗. (1.61)

Ademas, en la variable nueva (1.59), la funcion salida esta dada por

y(t) = Bz(t) = B1A−1/20

( iz2(t) − iz1(t)

2

)+B2

(z1(t) + z2(t)

2

). (1.62)

Recordando las hipoteses sobre B1 y B2 en (1.58), el operador B pertenence a L(D(A), Y ).

Por lo siguiente, suponemos que sistema (1.55)–(1.57) es observable. Como una conse-

cuencia directa de eso obtenemos que sistema (1.60)–(1.62) es observable y la estimacion

de la resolvente (1.33) se verifica.

Ahora introducimos los esquemas discretizados en tiempo que nos interesan. Para todo

t > 0 y β > 0, consideramos el esquema de Newmark para el sistema (1.55):

uk+1 + uk−1 − 2uk

(t)2 +A0

(βuk+1 + (1 − 2β)uk + βuk−1

)= 0,

(u0 + u1

2,u1 − u0

t)

= (u0, v0) ∈ D(A120 ) ×H.

(1.63)

La energıa de (1.63) es

Ek =

∥∥∥∥A1/20

(uk + uk+1

2

)∥∥∥∥2

+

∥∥∥∥uk+1 − uk

t

∥∥∥∥2

+ (4β − 1)(t)2

4

∥∥∥∥A1/20

(uk+1 − uk

t)∥∥∥∥

2

, k ∈ Z, (1.64)

la cual es analoga a la energıa del modelo continuo (1.56). Multiplicando la primera ecuacion

del sistema (1.63) por (uk+1 − uk−1)/2t, tomando sus normas e integrando por partes,

vemos facilmente que (1.64) queda constante con respecto de k. Ademas, por lo siguiente,

suponemos que β ≥ 1/4 para garantizar que la sistema (1.63) es estable incondicionalmente.

La funcion salida se da por la discretizacion de (1.57), como sigue:

yk = B1

(uk + uk+1

2

)+B2

(uk+1 − uk

t), (1.65)

donde suponemos que B1 o B2 se anulan.

Enfatizamos que, para todo s > 0, el espacio del filtrado Cs en (1.38) definido ahora

como

Cs = span Ψj : donde sus autovalores iµj satisfacen |µj| ≤ s (1.66)


tiene su sentido, si Ψjj∈N∗ satisfacen (1.61). Notese que este espacio es invariante bajo

la accion del semi-grupo discreto asociado al esquema de Newmark discretizado en tiempo

(1.63).

Consecuentemente, demostramos el siguiente Teorema:

Theorem 1.3.4. Dado β ≥ 1/4 y δ > 0. supongamos que B1 ≡ 0 o B2 ≡ 0. Entonces existe

un tiempo Tδ tal que para cualquier T > Tδ, existe una constante positiva kT,δ, tal que para

t suficiente pequena, la solucion de (1.63) con dato inicial (u0, v0) ∈ Cδ/t satisface

kT,δE1/2 ≤ t

∑

kt∈(0,T )

∥∥∥yk∥∥∥

2

Y, (1.67)

donde yk esta definida en (1.65) y B1, B2 satisfacen (1.58).

Ademas, Tδ puede elegirse como

Tδ,1 = π[(1 + βδ2)2

(1 +

(β − 1

4)δ2)2M2 +m2C2

B,1

δ

16

4]1/2, (1.68)

cuando B2 = 0 y como

Tδ,2 = π[(1 + βδ2)2

(1 +

(β − 1

4

)δ2)M2 +m2C2

B,2

δ4

16

]1/2, (1.69)

cuando B1 = 0.

Una de las aplicaciones interesantes de nuestros resultados del Capıtulo 5 es que se per-

mite proponer una estrategia de dos paso para analizar la observabilidad de la aproximacion

discretizada completa de (1.29)-(1.30). En el primero paso, consideramos la propiedad de

observabilidad del esquema discretizado en espacio, como se viene haciendo en muchos otros

casos ([5], [12], [13], [32], [75], [108] y [110] para mas referencias). En el segundo paso, usando

la observabilidad uniforme con respecto de la malla de espacio, deducimos la desigualdad

uniforme de observabilidad para los esquemas discretizados completamente (tanto en es-

pacio como en tiempo), a partir del resultado en este capıtulo. Veasen los detalles de la

aplicacıon en el Capıtulo 5.

Notese que el tiempo de observabilidad Tδ en nuestro resultados no es optimo. En rea-

lidad, especialmente en el caso continuo, el tiempo optimo de observabilidad que se obtiene

por estimaciones de la resolvente esta muy lejos del que se obtiene por la Optica Geometrica.

Notese tambien que el Teorema 1.3.4 tambien se puede aplicar a diversas aplicaciones

relevantes. Especialmente, los resultados sobre la observabilidad de la frontera de la ecuacion


de ondas dicretizada en tiempo del Capıtulo 4, se pueden comprender como un caso par-

ticular. Cuando aplicando este resultado al sistema (1.18), la condicion de pequenez δ se

puede eliminar, pero sin mantener la relacion optima entre el parametro de filtrado δ y el

tiempo optimo de observabilidad Tδ.

El resto de la Tesis esta organizando como sigue. En el Capıtulo 2, presentamos proble-

mas y herramientas basicas de la teorıa para los sistemas de dimension finita, aquellos que

se han utilizado recientemente en el contexto de EDPs y su analisis numerico. Ademas, in-

troducimos los modelos del caso continuo y unos resultados preliminares. Los Capıtulos 3–5

son el nucleo de la Tesis. Cada uno de ellos contiene los detalles de los resultados descritos

en las Secciones 1.1–1.3. En el ultimo capıtulo de la Tesis presentamos un resumen breve

de los resultados ademas de una lista de problemas abiertos y de lıneas de investigacion

futuras en esta tematica.

Abstract

In this Thesis we analyze various time-discrete schemes for the heat, wave and conser-

vative linear systems. Our main goal is to study how the time discretization of the partial

differential equations under consideration affect the well-known properties of the continuous

models, as observability, controllability, and so on.

For the heat equation, we study the null-controllability of an implicit Euler time-discrete

scheme in a bounded domain with a local internal controller. Using Lebeau-Robbiano’s time

iteration method, we prove that the projection of the solutions in an appropriate filtered

space is null controllable with uniformly bounded controls with respect to the time-step

parameter. In this way, the well-known null-controllability property of the time-continuous

heat equation can be proven as the limit, as the time-step parameter tends to zero, of the

controllability of projections of the time-continuous one.

Analogous results can also be obtained for other discrete schemes, as the explicit Euler

scheme, the θ-method, etc. We also discuss the null-controllability of the implicit Euler time-

discrete parabolic equation of fractional order, generalizing the results in the continuous

setting.

For the wave equation, we study the exact boundary controllability of a trapezoidal

time-discretization in a bounded domain. We prove, using multiplier techniques, that the

projection of the solution in an appropriate filtered space is exactly controllable with uni-

formly bounded cost with respect to the time-step. In this way, the well-known exact

controllability property of the time-continuous wave equation can be reproduced as the

limit, as the time-step parameter tends to zero, of the controllability of projections of the

time-discrete one. By duality these results are equivalent to deriving uniform observability

estimates (with respect to the time-step) within a class of solutions of the time-discrete

problem in which the high frequency components have been filtered. The later can be

established by means of a time-discrete version of the classical multiplier technique. The

optimality of the order of the filtering parameter can also be established, although a careful

analysis of the expected velocity of propagation of time-discrete waves indicates that its

exact value could be improved.

The last result of this Thesis concerns the uniform observability of various time-discrete

schemes of an abstract evolution system zt = Az, where A is a skew-adjoint operator, and an

observation operator B is given. More precisely, we assume that the pair (A,B) is exactly

observable in the continuous setting, and we derive uniform observability inequalities for

suitable time-discrete schemes within the class of conveniently filtered initial data. The

method we use is mainly based on the so called resolvent estimate. We present some

applications of our results to time-discrete schemes for wave equations, KdV equations and

fully discrete approximations schemes for wave equations. In particular, the previous results

on uniform boundary observability of the time-discrete wave equation can be seen as a direct

consequence of this abstract results.

Chapter 1

Introduction and main results

Our main goal of this Thesis is to analyze control problems for numerical approximation

schemes for diffusion and wave propagation processes. We mainly focus on time-discrete

schemes. More precisely, we study how the time discretization of the partial differential

equations under consideration affect the well-known properties of the continuous models,

as observability, controllability, and so on.

Control Theory originated and has been strongly inspired by industrial applications such

as airplanes, chemical plants, space vehicles and so on (see, for instance, [20], [110]). The

foundations of finite-dimensional control theory were established by R. E. Kalman ([41],[42])

and since then, the theory has been greatly generalized first to linear and nonlinear infinite

dimensional systems and to stochastic systems, etc.([11], [15], [55], [57], [63], [87], [90] and

the references therein).

Control theory addresses a variety of different problems. One of the most natural and

significant ones is that of controllability. It can be formulated, roughly, as follows. Consider

an evolution system (either described in terms of Partial or Ordinary Differential Equations

(PDE/ODE)). We are allowed to act on the trajectories of the system by means of a suitable

control (the right hand side of the system, the boundary conditions, etc.). Then, for a given

time interval t ∈ (0, T ) and initial and final states, we have to find a control such that the

solution matches both the initial state at time t = 0 and the final one at time t = T . There

is a large literature on these topics. We refer for instance to the book by Lee and Marcus

[53] for an introduction of those problems in the context of finite-dimensional systems. We

also refer to the survey paper by Russell [87] and to the book of Lions [55] (also to his survey

paper [57]) for an introduction to the controllability of PDEs, and to the more recent works

[66], [107], [109] and [112].

1

2 CHAPTER 1. INTRODUCTION AND MAIN RESULTS

In 1988, Lions ([57]) introduced the so called Hilbert Uniqueness Method (shorted as

HUM). Roughly speaking, it is based on the principle that, whenever a system is control-

lable, the control can be built by minimizing a suitable quadratic functional defined on the

class of solutions of the adjoint system. Suitable variants of this functional allow building

controls of minimal L2-norm, bang-bang controls, approximate controls, etc. The main

difficulty when minimizing these functionals is to show that they are coercive. This turns

out to be equivalent to the so called observability property for the adjoint equation, which

provides global estimates on the adjoint state everywhere in terms of partial measurements.

For instance, one of the most typical problem is that of boundary observability in which

the goal is to evaluate the energy of the whole system via boundary measurements.

Controllability/Observability of continuous systems is well understood for scalar linear

models. However, to build efficient numerical approximation schemes one necessarily needs

to deal with similar issues for suitable discretization schemes. But, it is by now well-known

that discretizations may destroy the controllability/observability properties of continuous

systems. This happens, for instance, in the context of wave propagation since the interaction

of waves with a numerical mesh produces dispersion phenomena and spurious high frequency

oscillations ([94], [97]). In particular, because of this nonphysical interaction of waves with

the discrete medium, the velocity of propagation of numerical waves, the so called group

velocity, may converge to zero when the wavelength of solutions is of the order of the size

of the mesh and the latter tends to zero. As a consequence of this fact, the time needed to

uniformly (with respect to the mesh size) observe (or control) the numerical waves from the

boundary or from a subset of the medium in which they propagate may tend to infinity as

the mesh becomes finer. Thus, the observation and control properties of the discrete model

may fail to be uniform with respect to the mesh-size parameters.

The analysis of controllability/observability properties of numerical approximation sche-

mes has been the object of intensive studies even though most analytical results concern

the case of space semi-discretizations ([5], [25], [32], [60], [73], [106] and [110]). This Thesis

is devoted to analyze time discretization schemes and their controllability/observability

properties. The goal is to develop a theory allowing to understand which time discretization

schemes provide uniform controllability/observability properties and, when this fails to be

the case, how uniformity may be reestablished by suitably relaxing these two problems.

More precisely, in this Thesis we address the following three problems:

1. Interior control of the time-discrete heat equation.

1.1. CONTROLLABILITY OF THE TIME-DISCRETE HEAT EQUATION 3

2. Boundary controllability and observability of the time-discrete wave equation.

3. Observability of linear time-discrete conservative systems.

We now describe more precisely the problems studied in this Thesis, our main results

and the methods we have developed.

1.1. Controllability of the time-discrete heat equation

Chapter 3 is devoted to the internal controllability of the time-discrete heat equation in

a bounded domain.

Let Ω be a given open bounded domain in lRd (d ∈ lN∗) with C∞ boundary ∂Ω = Γ,

and ω be a given non-empty open subset of Ω. We consider the following heat equation

with local internal controller:

yt − ∆y = u1ω, (t, x) ∈ (0, T ) × Ω

y = 0, (t, x) ∈ (0, T ) × Γ

y(0, x) = y0(x), x ∈ Ω.

(1.1)

It is well-known that system (1.1) is null controllable1 for any T > 0 and for any non-

empty open subset ω ⊂ Ω (see, for instance, [23] and [49]), i.e. for any y0 ∈ L2(Ω) there

exists a control u ∈ L2((0, T ) × ω), such that the corresponding solution of (1.1) satisfies

y(T, x) = 0, ∀ x ∈ Ω.

Moreover, one has the following estimate for the minimal L2-norm null control u of system

(1.1):

‖u‖L2((0,T )×ω) ≤ C∥∥y0∥∥

L2(Ω),

where C is a positive constant depending only on T,Ω and ω. Here and thereafter, we will

use C to denote a generic positive constant (independent of the time or space discretization

parameters) which may vary from line to line.

For any given K ∈ lN∗, we set t = T/K and introduce the net

t0 = 0 < t1 < · · · < tK = T

1The approximate controllability property can be formulated similarly, i.e., for any y0, y1 ∈ L2(Ω) and

ε > 0, we look for a control u ∈ L2((0, T )×ω) such that the solution of (1.1) satisfies‚

‚y(T ) − y1‚

‚

L2(Ω)≤ ε.


with tk = kt and k = 0, 1, · · · ,K.

The time-discrete version of system (1.1) reads:

yk+1 − yk

t − ∆yk+1 = uk1ω, x ∈ Ω, k = 0, 1, · · · ,K − 1

yk = 0, x ∈ Γ, k = 1, · · · ,Ky0 ∈ L2(Ω) given.

(1.2)

System (1.2) is an implicit time-discrete approximation of the heat equation with control on

the subset ω ⊂ Ω. Here yk0≤k≤K stands for the state and uk0≤k≤K−1 for the control.

Let us introduce the property of null-controllability2 for this time-discrete system: Sys-

tem (1.2) is said to be null controllable in time T (for any given t > 0) if for any

y0 ∈ L2(Ω) there exists a control uk ∈ L2(ω)0≤k≤K−1 with K = T/t, called henceforth

a discrete null control, such that the solution ykk=0,··· ,K of (3.1) satisfies

yK(x) = 0, ∀ x ∈ Ω.

The first result for the control system (1.2) is that, whatever Ω ⊂ lRd is, this null-

controllability property of system (1.2) fails:

Theorem 1.1.1. Let ω be a nonempty subset of Ω. For any given t > 0, system (1.2) is

neither null nor approximately controllable.

In view of the lack of controllability for system (1.2) it is natural to relax the con-

trollability requirement by considering the projections of solutions over a suitable class of

low frequency Fourier components. For this reason, we introduce the following classes of

functions in which the high frequency components have been filtered through a Fourier

decomposition argument.

Let Φjj≥1 ⊂ H10 (Ω) be an orthonormal basis of L2(Ω) consisting on the eigenvectors

of the Dirichlet Laplacian:

−∆Φj = µ2jΦj, x ∈ Ω

Φj = 0, x ∈ Γ.(1.3)

For any s > 0, define

C1,s = f(x) | f(x) =∑

µ2j<s

ajΦj(x), aj ∈ lC ⊂ H10 (Ω), (1.4)

2We can define similarly the approximate controllability property of the time-discrete system as follows:

System (1.2) is said to be approximately controllable in time T if for any y0 ∈ L2(Ω), any final state

yT ∈ L2(Ω) and ε > 0, there exists a control uk ∈ L2(ω)0≤k≤K−1 such that the solution ykk=0,··· ,K of

(1.2) satisfies‚

‚yK − yT‚

‚

L2(Ω)< ε.

1.1. CONTROLLABILITY OF THE TIME-DISCRETE HEAT EQUATION 5

C0,s = g(x) | g(x) =∑

µ2j<s

bjΦj(x), bj ∈ lC ⊂ L2(Ω), (1.5)

and

C−1,s = z(x) | z(x) =∑

µ2j<s

cjΦj(x), cj ∈ lC ⊂ H−1(Ω), (1.6)

subspaces of H10 (Ω), L2(Ω) and H−1(Ω), respectively, with the induced topologies. It is

clear that

∞⋃

k=1

C1,k is dense in H10 (Ω), and the same can be said for

∞⋃

k=1

C0,k in L2(Ω) and

∞⋃

k=1

C−1,k in H−1(Ω). Denote by π1,s, π0,s and π−1,s the projection operators from H10 (Ω),

L2(Ω) and H−1(Ω) to C1,s, C0,s and C−1,s, respectively. The spaces C±1,s and the projectors

π±1,s will not be used in this section but we will need them later.

The second result for system (1.2) shows that the projection of the solution of system

(1.2) on C0,s is uniformly null controllable, with an appropriate choice of the value of the

filtering parameter s as a function of the dicretization parameter t:

Theorem 1.1.2. For any fixed T > 0 and r ∈ (0, 2), there exists a positive constant

Λ = Λ(r, T,Ω, ω) such that for all y0 ∈ L2(Ω), there exists a control uk ∈ L2(ω)k=0,··· ,K−1,

so that

(1) The solution of system (1.2) satisfies

π0,Λ(t)−ryK(x) = 0, ∀ x ∈ Ω; (1.7)

(2) There exists a constant C = C(r, T,Ω, ω) > 0, independent of t, such that

tK−1∑

k=0

∫

ω|uk|2dx ≤ C

∫

Ω|y0|2dx (1.8)

holds for any t > 0 and y0 ∈ L2(Ω).

The proof of Theorem 1.1.2 is based on Lebeau-Robbiano’s time iteration method in-

troduced in [49] that we adapt here to the time-discrete setting (see the details in Section

3.4).

Moreover, we have the following convergence property and error estimate for the discrete

null controls:


Theorem 1.1.3. For the discrete null control uk0≤k≤K−1 given in the proof of Theorem

1.1.2, it holds

UK(·, x) =

K−1∑

k=0

uk(x)1[tk ,tk+1)(·) −→ u(·, x) strongly in L2((0, T ) × ω) as t→ 0,

where u is a null control of system (1.1). Moreover, there exists a constant C > 0, inde-

pendent of t and y0, such that UK and u satisfy

∥∥UK − u∥∥

L2((0,T )×ω)≤ C

√t∥∥y0∥∥

L2(Ω). (1.9)

In the last part of Chapter 3 we discuss some other time-discrete schemes and the

corresponding controllability properties for the control heat equation (1.1), for instance,

the implicit Euler method, the θ-method, etc. Similarly, we analyze the time implicit time-

discrete fractional order parabolic system and obtain similar results.

1.2. Controllability of the time-discrete wave equation

In Chapter 4, we study the exact boundary controllability properties for the trapezoidal

time-discrete wave equation in a bounded domain.

Let Ω be given as before, Γ0 a nonempty open set of Γ, and T > 0 a given time duration.

We consider the following controlled (time continuous) wave equation with a controller

acting on the subset Γ0 of the boundary:

ytt − ∆y = 0 in (0, T ) × Ω

y = u1Γ0 on (0, T ) × Γ

y(0) = y0, yt(0) = y1 in Ω.

(1.10)

In (1.10), (y(t, ·), yt(t, ·)) is the state and u(t, ·) is the control. The state and control

spaces for system (1.10) are chosen to be L2(Ω)×H−1(Ω) and L2((0, T )×Γ0), respectively.

The property of exact (boundary) controllability of (1.10) is defined as follows: For any

(y0, y1) ∈ L2(Ω)×H−1(Ω), there exists a control u ∈ L2((0, T )× Γ0) such that the solution

y ∈ C([0, T ];L2(Ω)) ∩ C1([0, T ];H−1(Ω)) of (1.10), defined by the classical transposition

method ([57]), satisfies:

y(T ) = yt(T ) = 0 in Ω. (1.11)

Unlike in the context of heat equations, the controllability property for the wave equation

(1.10) holds only under suitable geometric restrictions on the subset Γ0 of the boundary

where the control acts and provided that the controllability time T is large enough.

1.2. CONTROLLABILITY OF THE TIME-DISCRETE WAVE EQUATION 7

By classical duality arguments ([57]), the above controllability property is equivalent to

a (boundary) observability estimate for the following uncontrolled wave equation:

ϕtt − ∆ϕ = 0, in (0, T ) × Ω

ϕ = 0 on (0, T ) × Γ

ϕ(T ) = ϕ0, ϕt(T ) = ϕ1, in Ω.

(1.12)

The desired observability inequality reads as follows:

E(0) ≤ C

∫ T

0

∫

Γ0

∣∣∣∂ϕ

∂ν

∣∣∣2dΓ0dt, ∀ (ϕ0, ϕ1) ∈ H1

0 (Ω) × L2(Ω). (1.13)

On the other hand, E(0) stands for the energy E(t) of (1.12) at t = 0, with

E(t) =1

2

∫

Ω

[|ϕt(t, x)|2 + |∇ϕ(t, x)|2

]dx, (1.14)

which is constant in time.

Inequality (1.13) asserts that the total energy of any solution of (1.12) can be observed

in terms of the energy concentrated on Γ0 in the time interval (0, T ). It is well-known that

there are typically two classes of conditions on (T,Ω,Γ0) guaranteeing (1.13). The first one

is given by the classical multiplier condition. More precisely, fix some x0 ∈ lRd, put

Γ0=x ∈ Γ

∣∣ (x− x0) · ν(x) > 0, R

= max

x∈Ω|x− x0|, (1.15)

where ν(x) is the unit outward normal vector of Ω at x ∈ Γ. Then (1.13) holds for Γ0 as

in (1.15) provided T > 2R. This is the typical situation one encounters when applying the

multiplier technique ([57]), and Carleman inequalities (e.g. [102]) to deduce (1.13). The

second one is when (T,Ω,Γ0) satisfy the Geometric Control Condition (GCC, for short)

introduced in [2], which asserts that all rays of geometric optics in Ω intersect the sub-

boundary Γ0 in a uniform time T . In this case, (2.14) is established by means of tools from

micro-local analysis ([2]). This condition is optimal.

We then introduce the following trapezoidal time semi-discretization of (1.10):

yk+1 + yk−1 − 2yk

(t)2 − ∆(yk+1 + yk−1

2

)= 0, in Ω, k = 1, · · · ,K − 1

yk = uk1Γ0 , on Γ, k = 0, · · · ,Ky0 = y0, y1 = y0 + ty1.

(1.16)

Here (y0, y1) ∈ L2(Ω)×H−1(Ω) are the data given in system (1.10) that allow determin-

ing the initial data for the time-discrete system too. The well-posedness of system (1.16)

can be proved by means of the transposition method.


The controllability problem for system (1.16) may be formulated as follows: For any

(y0, y1) ∈ L2(Ω)×H−1(Ω), to find a control uk ∈ L2(Γ0)k=1,··· ,K−1 such that the solution

ykk=0,··· ,K of (1.16) satisfies:

yK−1 = yK = 0 in Ω. (1.17)

Note that (1.17) is equivalent to the condition yK−1 = (yK−yK−1)/t = 0 that is a natural

discrete version of (1.11).

As in the context of the above continuous wave equation, we also consider the uncon-

trolled system

ϕk+1 + ϕk−1 − 2ϕk

(t)2 − ∆

(ϕk+1 + ϕk−1

2

)= 0, in Ω, k = 1, · · · ,K − 1

ϕk = 0, on Γ, k = 0, · · · ,KϕK = ϕt

0 + tϕt1 , ϕK−1 = ϕt

0 , in Ω,

(1.18)

where (ϕt0 , ϕt

1 ) ∈ (H10 (Ω))2. In particular, to guarantee the convergence of the solutions

of (1.18) towards those of (1.12) we assume that

ϕt

0 → ϕ0 strongly in H10 (Ω),

ϕt1 → ϕ1 strongly in L2(Ω),

as K → ∞, or t→ 0, (1.19)

with tϕt1 being bounded in H1

0 (Ω). Obviously because of the density of H10 (Ω) in L2(Ω)

this choice is always possible.

The energy of system (1.18) is given by

Ekt

=

1

2

∫

Ω

(∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

+|∇ϕk+1|2 + |∇ϕk|2

2

)dx, k = 0, · · · ,K − 1, (1.20)

which is a discrete counterpart of the continuous energy E(t) in (1.14). Multiplying the

first equation of system (1.18) by (ϕk+1−ϕk−1)/2 and integrating it in Ω, using integration

by parts, it is easy to show the following property of conservation of energy:

Ekt = E0

t, k = 0, · · · ,K − 1. (1.21)

Consequently the scheme under consideration is stable and its convergence (in the classical

sense of numerical analysis) is guaranteed in an appropriate functional setting (in particular

in the finite-energy space H10 (Ω) × L2(Ω), under the condition (1.19)).

1.2. CONTROLLABILITY OF THE TIME-DISCRETE WAVE EQUATION 9

By means of classical duality arguments, it is easy to show that the above controllability

property (1.17) is equivalent to the following boundary observability property for solutions

ϕkk=0,··· ,K of (1.18):

E0t ≤ Ct

K−1∑

k=1

∫

Γ0

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ0, ∀ (ϕt0 , ϕt

1 ) ∈ (H10 (Ω))2. (1.22)

Similar to Theorem 1.1.1, we have the following negative result for systems (1.16)-(1.18):

Theorem 1.2.1. For any given t > 0 and any nonempty open subset Γ∗ of Γ, system

(1.18) is not observable, and therefore, system (1.16) is not null controllable.

The uniform observability of system (1.18) requires of a suitable filtering of the high

frequency components of solutions and can be stated as follows:

Theorem 1.2.2. Let T > 2R. Then there exist two constants h0 > 0 and δ > 0, depending

only on d, T and R, such that for all (ϕt0 , ϕt

1 ) ∈ C1,δ(t)−2 ×C0,δ(t)−2 , the corresponding

solution ϕkk=0,··· ,K of (1.18) satisfies

E0t ≤ Ct

K−1∑

k=1

∫

Γ0

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ0, (1.23)

for all t ∈ (0, h0].

The key ingredient in the proof of Theorem 1.2.2 is a time-discrete multiplier method,

that we develop in Section 4.6.

At this point it is important to indicate that the reverse of (1.23) is true for all T > 0,

uniformly on t > 0, without the need of any filtering of the high frequencies. As an

immediate consequence of this, by transposition, one derives the (uniform with respect to

t > 0) well-posedness of (1.16).

As a consequence of Theorem 1.2.2, we have the following uniform controllability result

for system (1.16) and the convergence result for the controls :

Theorem 1.2.3. Let T , h0 and δ be given as in Theorem 1.2.2, and K > 1 be an odd

integer. Then for any t ∈ (0, h0] and any (y0, y1) ∈ L2(Ω) × H−1(Ω), there exists a

control uk ∈ L2(Γ0)k=1,··· ,K−1 such that the solution of (1.16) satisfies

π0,δ(t)−2yK−1 = π−1,δ(t)−2

(yK − yK−1

t)

= 0 in Ω; (1.24)


There exists a constant C > 0, independent of t, y0 and y1, such that

tK−1∑

k=1

∥∥∥uk∥∥∥

2

L2(Γ0)≤ C

∥∥∥∥(y0,

y1 − y0

t

)∥∥∥∥2

L2(Ω)×H−1(Ω)

; (1.25)

When t→ 0,

Ut=

K−1∑

k=1

uk(x)1[kt,(k+1)t)(t) −→ u strongly in L2((0, T ) × Γ0), (1.26)

where u is a control of system (1.10), fulfilling (1.11);

When t→ 0,

yt= y010(t) +

1

t

K−1∑

k=0

[(t− kt)yk+1 −

(t− (k + 1)t

)yk]1(kt,(k+1)t](t)

−→ y strongly in C([0, T ];L2(Ω)) ∩H1([0, T ];H−1(Ω)),

(1.27)

where y is the solution of system (1.10) with the limit control u as above.

Further, we show the following optimality result, which means that the order (t)−2 of

the filtering parameter in Theorem 1.2.2 is sharp.

Theorem 1.2.4. Assume Γ∗ is any nonempty open subset of Γ. Then, for any given a > 2,

it follows that

limt→0

sup(ϕt

0 ,ϕt1 )∈C1,(t)−a×C0,(t)−a

E0t

tK−1∑

k=1

∫

Γ∗

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ∗

= ∞. (1.28)

We also analyze this issue from the microlocal point of view by analyzing the velocity

of propagation of singularities along bicharacteristic rays. This analysis confirms the opti-

mality of the order in the filtering parameter although it also shows that the value δ of the

filtering parameter that Theorem 1.2.2 yields is not the optimal one.

In this chapter we deal with the problem of boundary control, instead of that of internal

control of the previous chapter. In fact, as a consequence of the results on boundary control

in this chapter one could easily get the results for the case where the control acts on a

neighborhood of a subset of the boundary of the form Γ0.

1.3. OBSERVABILITY OF TIME-DISCRETE CONSERVATIVE LINEAR SYSTEMS11

1.3. Observability of time-discrete conservative linear sys-

tems

Chapter 5 is devoted to the analysis of the uniform observability property of abstract

time-discrete conservative linear systems.

Let X be a Hilbert space endowed with the norm ‖·‖X and A : D(A) → X a skew-adjoint

operator with compact resolvent. Let us consider the following abstract system:

zt(t) = Az(t), z(0) = z0. (1.29)

In (1.29), z0 ∈ X is the initial state, z = z(t) is the state. Such systems are often

used as models of vibrating systems (e.g., the wave equation), electromagnetic phenom-

ena (Maxwell’s equations) or in quantum mechanics (Schrodinger’s equation).

Assume that Y is another Hilbert space equipped with the norm ‖·‖Y . We denote by

L(X,Y ) the space of bounded linear operators from X to Y , endowed with the classical

norm. Let B ∈ L(D(A), Y ) be an observation operator and define the output function

y(t) = Bz(t). (1.30)

In order to give a sense to (1.30), we make the assumption that B is an admissible

observation operator in the following sense (see [98]): B is said to be admissible if for every

T > 0, there exists a constant KT ≥ 0 such that

∫ T

0‖y(t)‖2

Y dt ≤ KT ‖z0‖2X ∀ z0 ∈ D(A). (1.31)

Note that if B is bounded, i.e. if it can be extended such that B ∈ L(X,Y ), then B is

clearly an admissible observation operator. However, in applications, this is often not the

case, and the admissibility condition is then a consequence of a suitable “hidden regularity”

property of the solutions of the evolution equation (1.29).

The exact observability property of system (1.29)-(1.30) can be formulated as follows:

System (1.29)-(1.30) is said to be exactly observable in time T if there exists kT ≥ 0 such

that

kT ‖z0‖2X ≤

∫ T

0‖y(t)‖2

Y dt ∀ z0 ∈ D(A). (1.32)

Moreover, (1.29)-(1.30) is said to be exactly observable if it is exactly observable in some

time T > 0.


Note that since Controllability and Observability are dual notions, each statement con-

cerning observability has its counterpart in controllability. In this chapter, we mainly focus

on the observability properties of (1.29)-(1.30).

During Chapter 5, we assume that the continuous system (1.29)-(1.30) is exactly observ-

able. Consequently, according to the Hautus-type test, the pair (A,B) fulfils a condition

that is independent of T ([3], [70]). More precisely, as it was proved in [70], under the exact

observability property, the following resolvent estimate holds:

There exist constants M,m > 0 such that

M2 ‖(iωI −A)z‖2 +m2 ‖Bz‖2Y ≥ ‖z‖2 , ∀ ω ∈ lR, z ∈ D(A).

(1.33)

Note that it is very natural to assume that the continuous system (1.29)–(1.30) is exactly

observable since there is an extensive literature providing results of that kind for wave,

plate, Schorodinger and elasticity equations, among other models and by various methods

including microlocal analysis, multipliers and Fourier series, etc. Our goal here is to develop

a theory allowing to get results for time-discrete systems as a direct consequence of those

corresponding to the time-continuous ones.

Now we consider the time semi-discretization of system (1.29)–(1.30). We are thus

replacing the continuous dynamics (1.29) and the corresponding observability inequality

(1.32) by the time-discrete ones.

We then introduce the following implicit midpoint time discretization of system (1.29):

zk+1 − zk

t = A(zk+1 + zk

2

), in X, k ∈ Z

z0 given.

(1.34)

The output function of (1.34) is given by

yk = Bzk, k ∈ Z. (1.35)

Note that (1.34)–(1.35) is the discrete version of (1.29)–(1.30).

Taking into account that the spectrum of A is purely imaginary, it is easy to show

that∥∥zk∥∥

Xis conserved in the discrete time variable k ∈ Z, i.e.

∥∥zk∥∥

X=∥∥z0∥∥

X. Conse-

quently the scheme under consideration is stable and its convergence (in the classical sense

of numerical analysis) is guaranteed in an appropriate functional setting.

The uniform exact observability problem for system (1.34) is formulated as follows: To

find a positive constant kT , independent of t, such that the solutions zk of system (1.34)


satisfy:

kT

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥Bzk∥∥∥

2

Y, (1.36)

to all initial data z0 in an appropriate class.

Clearly, (1.36) is a discrete version of (1.32). We are interested in understanding under

which assumptions (1.36) holds uniformly on t. One expects to do it so that, when letting

t→ 0, one recovers the observability property of the continuous model.

This also can be done by means of the spectral filtering mechanism as before. More

precisely, since A is skew-adjoint with compact resolvent, its spectrum is given by σ(A) =

iµj : j ∈ Λ with Λ = Z∗ or N∗ and where (µj)j∈Λ is a sequence of real numbers. Set

(Ψj)j∈Λ an orthonormal basis of eigenvectors of A associated to the eigenvalues (iµj)j∈Λ,

that is:

AΨj = iµjΨj . (1.37)

Hence, similarly as in (1.4)–(1.6), for any s > 0, we define

Cs = span Ψj : the corresponding iµj satisfies |µj| ≤ s. (1.38)

Note that the Hilbert space D(A) is endowed with the norm of the graph of A:

‖z‖21 = ‖z‖2

X + ‖Az‖2X .

The fact that B ∈ L(D(A), Y ) implies

‖Bz‖Y ≤ CBδ

t ‖z‖X , z ∈ Cδ/t, (1.39)

where CB is a positive constant independent of t.We will prove that inequality (1.36) holds uniformly (with respect to t > 0) in the

class Cδ/t for any δ > 0 and for Tδ large enough, depending on the filtering parameter δ.

This can be obtained as a direct consequence of the following theorem:

Theorem 1.3.1. Let δ > 0.

Assume that we have a sequence of vector spaces Xδ,t ⊂ X and a sequence of unbounded

operators (At, Bt) such that

(H1) For each t > 0, the operator At is skew-adjoint on Xδ,t, and the vector space

Xδ,t is globally invariant by At. Moreover,

‖Atz‖X ≤ δ

t ‖z‖X , ∀z ∈ Xδ,t, ∀t > 0. (1.40)


(H2) There exists a positive constant CB such that


(H3) There exist two positive constants M and m such that


Y ≥ ‖z‖2X ,

∀z ∈ Xδ,t ∪D(At),∀ω ∈ lR, ∀t > 0.(1.42)

Then there exists a time Tδ such that for all time T > Tδ, there exists a constant kT,δ such

that for t small enough, the solution of

zk+1 − zk

t = At

(zk+1 + zk

2

), in Xδ,t, k ∈ Z, . (1.43)

with initial data z0 ∈ Xδ,t satisfies

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)


2

Y, ∀ z0 ∈ Xδ,t. (1.44)

Moreover, comparing to the optimal observability time T0 = πM in the continuous setting,

Tδ can be taken to be such that

Tδ = π[(

1 +δ2

4

)2M2 +m2C2

B

δ4

16

]1/2, (1.45)

where CB is as in (1.39).

Theorem 1.3.1 will be proved in Chapter 5 and the key ingredients are the Fourier

transform analogue in the time-discrete level and the resolvent estimate.

Indeed, taking At = A, Bt = B and Xδ/t = Cδ/t obviously Theorem 1.3.1 provides

an observability result for system (5.1)–(5.2) within the class Cδ/t, which is stated as

follows:

Theorem 1.3.2. Assume that (A,B) satisfy (1.33) and that B ∈ L(D(A), Y ).

Then, for any δ > 0, there exist Tδ and t0 > 0 such that for any T > Tδ and

t ∈ (0,t0), there exists a positive constant kT,δ, independent of t, such that the solution

zk of (1.34) satisfies

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥Bzk∥∥∥

2

Y, ∀ z0 ∈ Cδ/t. (1.46)

Moreover, Tδ can be taken to be such that

Tδ = π[M2(1 +

δ2

4

)2+m2C2

B

δ4

16

]1/2, (1.47)



Theorem 1.3.1 is also useful to address observability issues for more general time-

discretization schemes of (1.29)-(1.30) other than (1.34). For instance, one can consider

time semi-discrete schemes the form

zk+1 = Ttzk, yk = Bzk, (1.48)

where Tt is a linear operator with the same eigenvectors as the operator A. We also

assume that the scheme under consideration is conservative in the sense that there exist

real numbers λj,t such that


Moreover, we assume that there is an explicit relation between λj,t and µj (as in (1.37))

of the following form:

λj,t =1

t h(µjt), (1.50)

where h : [−δ, δ] 7→ [−π, π] is a smooth strictly increasing function, i.e.

|h(η)| ≤ π, infh′(η), |η| ≤ δ > 0. (1.51)

Roughly speaking, the first part of (1.51) reflects the fact that one cannot measure frequen-

cies higher than π/t in a mesh of size t. Besides, the second part is a non-degeneracy

condition on the group velocity (see [94]) of solutions of (1.48) which is necessary to guar-

antee the propagation of solutions that is required for observability to hold.

We also assumeh(η)

η−→ 1 as η → 0, (1.52)

which guarantees the consistency of the time-discrete scheme with the continuous one (2.18).

In this mathematical setting, we will prove the second main result in Chapter 5, in which

we show that for any δ > 0, inequality (1.36) holds uniformly on t for solutions of (1.48)

when the initial data are taken in the class Cδ/t:

Theorem 1.3.3. Under assumptions (1.49), (1.50) and (1.51), for any δ > 0, there exists

a time Tδ such that for all T > Tδ, there exists a constant kT,δ > 0 such that for all tsmall enough, any solution of (5.34) with initial value z0 ∈ Cδ/t satisfies

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥∥B(zk + zk+1

2

)∥∥∥∥2

Y

. (1.53)


Besides, we have the following estimate on Tδ:

T 2δ < 2π2

[2 tan2

(h(δ)2

)+ C2

Bδ2m2

(1 + tan2

(h(δ)2

))

+ 2M2(

inf|ω|≤δ

∣∣∣(1 + tan2

(h(δ)2

))h′(ω)

∣∣∣)−2(

1 + tan4(h(δ)

2

))]. (1.54)


The proof of Theorem 1.3.3 is similar to the one of Theorem 1.3.1, see the details in

Section 5.3.1.

The last main result in Chapter 5 concerns a second order in time systems. Let H be

a Hilbert space endowed with the norm ‖·‖H and let A0 : D(A0) → H be a self-adjoint

positive operator with compact resolvent. We consider the initial value problem

utt +A0u = 0,

u(0) = u0, ut(0) = v0,(1.55)

which can be seen as a generic model for the free vibrations of elastic structures such as

strings, beams, membranes, plates or three-dimensional elastic bodies.

Of course, such systems can be written in the first-order form as (1.29). However, there

are time-discretization schemes such as the Newmark method which cannot be put in the

form (1.49). Hence it is natural to make a specific analysis of time-discretization schemes

for (1.55). We shall derive a natural extension of Theorem 1.3.1 to this class of second order

(in time) systems.

The energy of (1.55) is given by

E(t) = ‖ut(t)‖2H +

∥∥∥A1/20 u(t)

∥∥∥2

H, (1.56)


We consider the output function

y(t) = B1u(t) +B2ut(t), (1.57)

where B1 and B2 are two observation operators satisfying B1 ∈ L(D(A0), Y ) and B2 ∈L(D(A

1/20 ), Y ). In other words, we assume that there exists two constants CB,1 and CB,2,

such that

‖B1u‖Y ≤ CB,1 ‖A0u‖H , ‖B2v‖Y ≤ CB,2

∥∥∥A1/20 v

∥∥∥ . (1.58)

In the sequel, we assume either B1 = 0 or B2 = 0. Indeed, it is a necessary condition on

the time-discrete level, due to a technical problem that we cannot solve so far.


System (1.55)–(1.57) can be put in the form (1.29)–(1.30). Indeed, setting

z1(t) = ut + iA1/20 u, z2(t) = ut − iA

1/20 u, (1.59)

equation (1.55) is equivalent to

zt = Az, z =

(z1z2

), A =

(iA

1/20 0

0 −iA1/20

), (1.60)

for which the energy space is X = H × H with the domain D(A) = D(A1/20 ) × D(A

1/20 ).

Moreover, the energy E(t) given in (1.56) coincides with half of the norm of z in X.

Note that the spectrum of A is explicitly given by the spectrum of A0. Indeed, if (µ2j)j∈N∗

(µj > 0) is the sequence of eigenvalues of A0, i.e.


∗,

with corresponding eigenvectors φj , then the eigenvalues of A are ±iµj, with corresponding

eigenvectors

Ψj =

(φj

0

), Ψ−j =

(0

φj

), j ∈ lN∗. (1.61)

Besides, in the new variables (1.59), the output function is given by

y(t) = Bz(t) = B1A−1/20

( iz2(t) − iz1(t)

2

)+B2

(z1(t) + z2(t)

2

). (1.62)

Recalling the assumptions on B1 and B2 in (1.58), the admissible observation B belongs to

L(D(A), Y ).

In the sequel, we assume that the system (1.55)–(1.57) is exactly observable. As a direct

consequence of this we obtain that system (1.60)–(1.62) is exactly observable and therefore

the resolvent estimate (1.33) holds.

We now introduce the time-discrete schemes we are interested in. For any t > 0 and

β > 0, we consider the following Newmark time-discrete scheme for system (1.55):

uk+1 + uk−1 − 2uk

(t)2 +A0

(βuk+1 + (1 − 2β)uk + βuk−1

)= 0,

(u0 + u1

2,u1 − u0

t)

= (u0, v0) ∈ D(A120 ) ×H.

(1.63)


Ek+1/2 =

∥∥∥∥A1/20

(uk + uk+1

2

)∥∥∥∥2

+

∥∥∥∥uk+1 − uk

t

∥∥∥∥2

+ (4β − 1)(t)2

4

∥∥∥∥A1/20

(uk+1 − uk

t)∥∥∥∥

2

, k ∈ Z, (1.64)


which is a discrete counterpart of the continuous energy (1.56). Multiplying the first equa-

tion of (1.63) by (uk+1 − uk−1)/2t, taking its norm and using integration by parts, it is

easy to show that (1.64) remains constant with respect to k. Furthermore, we assume in

the sequel that β ≥ 1/4 to guarantee that system (1.63) is unconditionally stable.

The output function is given by the following discretization of (1.57):

yk+1/2 = B1

(uk + uk+1

2

)+B2

(uk+1 − uk

t), (1.65)

where, as in (1.57), we assume that either B1 or B2 vanishes.

Especially, we emphasize that, for any s > 0, the filtering space Cs in (1.38)

Cs = span Ψj : the corresponding iµj satisfies |µj | ≤ s (1.66)

makes sense, if Ψjj∈N∗ is given by (1.61).

Note that this space is invariant under the actions of the discrete semi-groups associated

to the Newmark time-discrete schemes (1.63).

Consequently, we prove by an ad-hoc treatment of these discretizations that we can

recover obtain for any δ > 0 the uniform observability inequality for solutions of (1.63)

with initial data in Cδ/t, under some restrictions on the observation operator B:

Theorem 1.3.4. Let β ≥ 1/4 and δ > 0. We assume that either B1 ≡ 0 or B2 ≡ 0.

Then there exists a time Tδ such that for all T > Tδ, there exists a positive constant

kT,δ, such that for t small enough, the solution of (1.63) with initial data (u0, v0) ∈ Cδ/t

satisfies

kT,δE1/2 ≤ t

∑

kt∈(0,T )

∥∥∥yk+1/2∥∥∥

2

Y, (1.67)

where yk+1/2 is defined in (1.65) and B1, B2 satisfy (1.58).

Besides, Tδ can be chosen as

Tδ,1 = π[(1 + βδ2)2

(1 +

(β − 1

4)δ2)2M2 +m2C2

B,1

δ

16

4]1/2, (1.68)

if B2 = 0 and as

Tδ,2 = π[(1 + βδ2)2

(1 +

(β − 1

4

)δ2)M2 +m2C2

B,2

δ4

16

]1/2, (1.69)

if B1 = 0.


One of the interesting applications of our results in Chapter 5 is that it allows us

to develop a two-step strategy to study the observability of fully discrete approximation

schemes of (1.29)-(1.30). First, one uses the observability properties for space semi-discrete

approximation schemes, uniformly with respect to the space mesh-size parameter, as it

has already been done in many cases (see [5], [12], [13], [32], [75], [108] and [110] for

more references). Second, from the results of this Chapter on time discretizations, the

observability inequality (with respect to the time and space mesh-sizes) for the fully discrete

approximation schemes is derived. The detailed application of this Theorem is given in

Chapter 5.

Note that the observability time Tδ in our results is not sharp. This is due to the fact,

even in the continuous level, the optimal observability time one gets by means of resolvent

estimates is far from being the sharp one that Geometric Optics yields.

Note that Theorem 1.3.4 can also be applied to deal with various relevant applications.

Especially, the results on the boundary observability of the time-discrete wave equation in

Chapter 4, can be seen as a particular case. When applying this result on system (1.18),

the smallness condition on δ can be eliminated, but without getting an optimal relation

between the filtering parameter δ and the optimal control time Tδ.

The rest of the Thesis is organized as follows. In Chapter 2, we present some of the basic

problems and tools of control theory for finite-dimensional systems that will be later used in

the context of PDEs and their numerical analysis. Moveover, we introduce the continuous

models that will be analyzed later, and also collect some preliminary results. Chapter 3–5

constitute the core of the Thesis. Each of them contains the detailed development of the

results we have described in the previous sections 1.1–1.3. In the last chapter of the Thesis

we present a brief summary of the main results and also a list of open problems and future

directions of research.

Chapter 2

Preliminaries

2.1. Introduction

The main goal of this chapter is to collect various basic concepts and well-known existing

results that will be used along this Thesis. In Section 2.2, we first introduce the notion of

controllability for finite dimensional systems. In Section 2.3 and 2.4, we give some concepts

and basic results on the control problem and related topics on heat and wave equations,

respectively. In the last section, we address a general model of conservative linear systems

and present a basic result in which we state the equivalence between several sufficient

conditions for the controllability property to hold.

2.2. Linear dynamical systems

The problem of controllability for a dynamical system can be formulated, roughly, as

follows: “Can any initial state of a given dynamical system be transferred to any desired

final state in a finite time by some control action?” If the answer is affirmative, we shall

say that the system is controllable. This notion was rigorously formulated by R. Kalman

([40], [41] and [42]) in the context of ordinary differential equations (ODEs). This section

is devoted to rigorously introduce this concept and recall the main result, which yields an

algebraic characterization of controllable systems.

Let n,m ∈ lN∗. We consider the following finite dimensional system:

x′(t) = Ax(t) +Bu(t), t ∈ (0, T )

x(0) = x0.(2.1)

In (2.1), A is a real n × n matrix, B a real n × m matrix and x0 a vector in lRn. The

function x : [0, T ] → lRn represents the state and u : [0, T ] → lRm the control.

21

22 CHAPTER 2. PRELIMINARIES

System (2.1) is said to be controllable in time T if every initial datum x0 ∈ lRn can be

driven to any final datum x1 ∈ lRn at time T .

Kalman introduced the following rank condition which turns out to be necessary and

sufficient for controllability: System (2.1) is controllable in some time T > 0 if and only if

rank[B,AB, · · · , An−1B] = n. (2.2)

Note that, according to this result, a system is controllable for some time T > 0 if and only

if it is controllable for all T > 0.

Now we discretize system (2.1) with respect to the time variable t as follows:

xk+1 − xk

t = Axk+1 +Buk+1, k = 0, · · · ,K − 1

x0 ∈ lRn given.(2.3)

The following result is also well-known:

Theorem 2.2.1. ([42])Assume that A,B satisfy the Kalman condition (2.2) and λ1, λ2,

· · · , λk, (k ≤ n) to be the distinct eigenvalues of A. Then system (2.3) is controllable if

t 6= 2νπi

λµ − λl, µ 6= l (2.4)

holds for any ν ∈ lN∗.

Remark 2.2.1. Note that, in particular, if A,B satisfy the Kalman rank condition (2.2),

the discrete system (2.3) is controllable for t sufficiently small. More precisely, (2.4) is

fulfilled when t satisfies

t < minµ6=l

∣∣∣2νπ

λµ − λl

∣∣∣. (2.5)

Remark 2.2.2. Note that, in Theorem 2.2.1, A is a finite dimensional operator. Assume

that Aσ is an approximation of the unbounded operator ∆ in 1-d, with the eigenvalues

λl = −l2, l = 1, · · · ,√σ. Thus, (2.5) can be rewritten as:

t < minµ6=l

∣∣∣2νπ

λµ − λl

∣∣∣ <∣∣∣

2π

σ − 1

∣∣∣. (2.6)

When Aσ contains more and more eigenvalues of ∆ the right side of (2.6) tends to zero.

This is in agreement with the fact that system (2.3) is not null controllable for any fixed

t > 0, if A is an unbounded operator. But it also shows that the controllability of the time-

continuous system can be recovered as a consequence of the controllability of time-discrete

one by letting t→ 0 and, simultaneously, σ2 → ∞.

2.3. THE HEAT EQUATION AND THE L-R METHOD 23

2.3. The heat equation and the L-R method

Let T > 0 be given, Ω be a given open bounded domain in lRd (d ∈ lN∗) with C∞

boundary ∂Ω, and ω a given non-empty open subset of Ω. Denote by 1ω the characteristic

function of ω.

We consider the following heat equation with local internal controller:

yt − ∆y = u1ω, (t, x) ∈ (0, T ) × Ω

y = 0, (t, x) ∈ (0, T ) × ∂Ω

y(0, x) = y0(x), x ∈ Ω.

(2.7)

In (2.7), y = y(t, x) is the state and u = u(t, x) is the control function acting on the subset

ω.

The following theorem is a consequence of classical results of existence and uniqueness of

solutions of nonhomogeneous evolution equations. All the details may be found, for instance

in [6].

Theorem 2.3.1. For any u ∈ L2((0, T ) × ω) and y0 ∈ L2(Ω), equation (2.7) has a unique

weak solution y ∈ C([0, T ], L2(Ω)) given by the variation of constants formula

y(t) = S(t)y0 +

∫ t

0S(t− s)u(s)1ωds

where (S(t))t∈lR is the semigroup of contractions generated by the heat operator in L2(Ω).

Moreover, if u ∈W 1,1((0, T )×ω) and y0 ∈ H2(Ω)∩H10 (Ω), equation (2.7) has a classical

solution y ∈ C1([0, T ], L2(Ω))∩C([0, T ],H2(Ω)×H10 (Ω)) and (2.7) is verified in L2(Ω) for

all t > 0.

Let us now introduce the notion of controllability. System (2.7) is said to be approx-

imately controllable in time T > 0 if for any initial state y0 ∈ L2(Ω), any final state

y1 ∈ L2(Ω) and any ε > 0, there exists a control u ∈ L2((0, T ) × ω) such that the solution

of (2.7) satisfies∥∥y(T ) − y1

∥∥L2(Ω)

≤ ε.

Simultaneously, system (2.7) is said to be null controllable in time T > 0 if for any y0 ∈L2(Ω) there exists a control u ∈ L2((0, T ) × ω), called henceforth a null control of (2.7),

such that the corresponding solution satisfies

y(T, x) = 0, ∀ x ∈ Ω.


It is well-known that system (2.7) is both approximately and null controllable for any

T > 0 and for any non-empty open subset ω ⊂ Ω (see, for instance, [23] and [49]). Moreover,

one has the following estimate for the minimal L2-norm null control u of system (2.7):

‖u‖L2((0,T )×ω) ≤ C∥∥y0∥∥

L2(Ω),

where C is a positive constant depending only on T,Ω and ω.

The null-controllability of system (2.7) is equivalent to an observability estimate for the

adjoint system:

−ϕt − ∆ϕ = 0, (t, x) ∈ (0, T ) × Ω

ϕ = 0, (t, x) ∈ (0, T ) × ∂Ω

ϕ(T, x) = ϕ0(x), x ∈ Ω.

(2.8)

More precisely, system (2.7) is null controllable if and only if there exists a positive constant

C > 0 such that

‖ϕ(0)‖L2(Ω) ≤ C ‖ϕ‖L2((0,T )×ω) (2.9)

holds for all solutions of (2.8) with initial data ϕ0 ∈ L2(Ω). Inequality (2.9) is the so called

“observability inequality” for the adjoint system (2.8).

In [49], Lebeau and Robbiano introduced a time iteration method to show the above

mentioned controllability result. A simplified presentation of this method was given by

Lebeau and Zuazua in [52] where the linear system of thermoelasticity was addressed. More

recently, Miller has applied it systematically to the analysis of other models: heat equations

([71]), Schrodinger equations ([69]) and the linear system of thermoelasticity ([72]). In the

sequel, we shall refer to it as the L-R method.

The L-R method is based on an observability estimate for the eigenfunctions of the

Dirichlet Laplacian, which is stated as follows:

Theorem 2.3.2. ([49], [52]) Let Ω be a bounded domain of class C∞. Let µjj≥1 and

Φjj≥1 be defined by system (1.3). Then for any open non-empty subset ω of Ω, there

exist two positive constants Cj(Ω, ω), j = 1, 2, such that∫

ω

∣∣∣∑

µ2j≤σ

ajΦj(x)∣∣∣2dx ≥ C1e

−C2√

σ∑

µ2j≤σ

|aj |2 (2.10)

holds for every σ > 0 and every sequence ajµ2j≤σ of complex numbers.

This result was implicitly used in more detail in [49] and proven in [52] by means of

Carleman inequalities.

2.3. THE HEAT EQUATION AND THE L-R METHOD 25

As a consequence of (2.10) one can prove that the observability inequality (2.9) holds for

solutions of system (2.8) with initial data in C0,σ, as defined in (1.5), with the observability

constant being of the order of exp(C√σ). This allows us to show that the projection of

solutions of (2.7) over C0,σ can be controlled to zero with a control of size exp(C√σ). Thus,

when controlling the frequencies with µ2j ≤ σ one increases the L2(Ω)-norm of the high

frequencies with µ2j ≥ σ by a multiplicative factor of the order of exp(C

√σ). However,

solutions of system (2.7) without control (f = 0) but with a vanishing projection of the

initial data over C0,σ, decay in L2(Ω) at a rate of the order of exp(−σt). This can be easily

seen by means of the Fourier-type series expansion of solutions. Thus, if we divide the time

interval [0, T ] into two parts [0, T/2] and [T/2, T ], we control to zero the frequencies µ2j ≤ σ

in the first interval and then let the equation evolve without control in the second interval,

it follows that, at time t = T , the projection of the solution u over C0,σ vanishes and the

norm of the high frequencies does not exceed the norm of the initial data u0.

This argument allows us to control to zero the projection of the solutions of (2.7) over

C0,σ for any σ > 0 but not the whole solution. For the later an iterative argument is

needed in which the interval [0, T ] has to be decomposed in a suitably chosen sequence of

subintervals [Tl, Tl+1] and the argument above needs to be applied in each subinterval to

control an increasing range of frequencies with µ2j ≤ σl and σl going to infinity at a suitable

rate. This iterative process leads to a L2-control that drives the solution to zero in time T .

We refer to [49] and [52] for more details in this respect.

It is important to underline that the strong dissipativity of the heat equation is essential

to make this argument work. Indeed, one needs to make sure that the dissipation rate is

stronger than the increase of the size of the controls as frequencies increase. This is so for

the heat equation where the dissipation rate is e−µ2j t while the increase of the controls, in

view of (2.10), is of the order of eC2µj .

Actually, the optimality of this kind of argument has been proven in [67] where it was

shown that the fractional order parabolic equation yt + (−∆)αy = 0 is null controllable for

α > 1/2 but that this fails to be true in the limiting case a = 1/2.


2.4. The wave equation: duality arguments

Let Ω and Γ0 be given as before. We consider the following controlled (time-continuous)

wave equation with a controller acting on the subset Γ0 of the boundary:

ytt − ∆y = 0 in (0, T ) × Ω

y = u1Γ0 on (0, T ) × Γ

y(0) = y0, yt(0) = y1 in Ω.

(2.11)

In (2.11), (y(t, ·), yt(t, ·)) is the state and u(t, ·) is the control. The state and control spaces

of system (2.11) are chosen to be L2(Ω) ×H−1(Ω) and L2((0, T ) × Γ0), respectively.

The following theorem is a consequence of the classical results of existence and unique-

ness of solutions of nonhomogeneous evolution equations. Full details may be found in

[59].

Theorem 2.4.1. For any u ∈ L2((0, T ) × Γ0) and (y0, y1) ∈ L2(Ω) × H−1(Ω) equation

(2.11) has a unique weak solution defined by transposition

(y, yt) ∈ C([0, T ], L2(Ω) ×H−1(Ω)).

Moreover, the map y0, y1, u → y, yt is linear and there exists C = C(T ) > 0 such that

‖(y, yt)‖L∞(0,T ;L2(Ω)×H−1(Ω)) ≤ C(‖(y0, y1)‖L2(Ω)×H−1(Ω) + ‖u‖L2((0,T )×Γ0)).

The property of exact (boundary) controllability of (2.11) is defined as follows: For any

(y0, y1) ∈ L2(Ω)×H−1(Ω), there exists a control u ∈ L2((0, T )× Γ0) such that the solution

y ∈ C([0, T ];L2(Ω)) ∩ C1([0, T ];H−1(Ω)) of (2.11), defined by the classical transposition

method ([57]), satisfies:

y(T ) = yt(T ) = 0 in Ω. (2.12)

This controllability property holds under suitable geometric restrictions on the subset Γ0 of

the boundary where the control acts and provided that the controllability time T is large

enough.

By classical duality arguments ([57]), the above controllability property is equivalent to

a (boundary) observability estimate of the following uncontrolled wave equation:

ϕtt − ∆ϕ = 0, in (0, T ) × Ω

ϕ = 0 on (0, T ) × Γ

ϕ(T ) = ϕ0, ϕt(T ) = ϕ1, in Ω.

(2.13)

2.5. CONSERVATIVE LINEAR SYSTEMS 27

The observability inequality reads as follows:

E(0) ≤ C

∫ T

0

∫

Γ0

∣∣∣∂ϕ

∂ν

∣∣∣2dΓ0dt, ∀ (ϕ0, ϕ1) ∈ H1

0 (Ω) × L2(Ω). (2.14)

On the other hand, E(0) stands for the energy E(t) of (2.13) at t = 0, with

E(t) =1

2

∫

Ω

[|ϕt(t, x)|2 + |∇ϕ(t, x)|2

]dx, (2.15)

which remains constant in time, i.e.

E(t) = E(0), ∀ t ∈ [0, T ]. (2.16)

Inequality (2.14) asserts that the total energy of any solution of (2.13) can be observed

in terms of the energy concentrated on Γ0 in the time interval (0, T ). It is well-known that

there are typically two classes of conditions on (T,Ω,Γ0) guaranteeing (2.14):

i) The first one is given by the classical multiplier condition. Fix some x0 ∈ lRd, put

Γ0=x ∈ Γ

∣∣ (x− x0) · ν(x) > 0, R

= max

x∈Ω|x− x0|, (2.17)

where ν(x) is the unit outward normal vector of Ω at x ∈ Γ. Then (2.14) holds for

Γ0 as in (2.17) provided T > 2R. This is the typical situation when applying the

multiplier technique ([57]), and Carleman inequalities (e.g. [102]) to deduce (2.14),

which can also be applied to many other models.

ii) The second one is when (T,Ω,Γ0) satisfy the Geometric Control Condition (GCC, for

short) introduced in [2], which asserts that all rays of Geometric Optics in Ω intersect

the sub-boundary Γ0 in a uniform time T . In this case, (2.14) is established by means

of tools from micro-local analysis ([2]). This condition is optimal.

2.5. Conservative linear systems

Let X be a Hilbert space endowed with the norm ‖·‖X and let A : D(A) → X be

a skew-adjoint operator with compact resolvent. Let us consider the following abstract

system:

zt(t) = Az(t), z(0) = z0. (2.18)

The element z0 ∈ X is called the initial state and z = z(t) is the state. Such systems

are often used as models of vibrating systems (e.g., the wave equation), electromagnetic

phenomena (Maxwell’s equations) or in quantum mechanics (Schrodinger’s equation).


Assume that Y is another Hilbert space equipped with the norm ‖·‖Y . We denote by

L(X,Y ) the space of bounded linear operators from X to Y , endowed with the classical

operator norm. Let B ∈ L(D(A), Y ) be an observation operator and define the output

function

y(t) = Bz(t). (2.19)

In order to give a sense to (2.19), we make the assumption that B is an admissible obser-

vation operator in the following sense (see [98]):

Definition 2.5.1. The operator B is an admissible observation operator for system (2.18)-

(2.19) if for every T > 0 there exists a constant KT > 0 such that

∫ T

0‖y(t)‖2

Y dt ≤ KT ‖z0‖2X ∀ z0 ∈ D(A). (2.20)

Note that if B is bounded in X, i.e. if it can be extended such that B ∈ L(X,Y ), then

B is obviously an admissible observation operator. In particular, there are very important

situations, as the boundary control/observation of the wave equation, where B fails to be

bounded but the admissibility condition holds because of the so-called hidden regularity

property.

Definition 2.5.2. System (2.18)-(2.19) is exactly observable in time T if there exists kT > 0

such that

kT ‖z0‖2X ≤

∫ T

0‖y(t)‖2

Y dt ∀ z0 ∈ D(A). (2.21)

System (2.18)-(2.19) is said to be exactly observable if it is exactly observable in some time

T > 0.

The following theorem is proved in [3] and [70]:

Theorem 2.5.1. Assume that A is a skew-adjoint operator in the Hilbert space X and that

B : D(A) → Y is an admissible observation operator. Then system (2.18)-(2.19) is exactly

observable if and only if there exists a constant δ > 0 such that

‖(iωI −A)z‖2 + ‖Bz‖2Y ≥ δ ‖z‖2 , ∀ ω ∈ lR, z ∈ D(A). (2.22)

This spectral condition can be viewed as a Hautus-type test, generalizing, in some sense,

the classical Kalman rank condition (see, for instance, [95]).

Moreover, since A is skew-adjoint with compact resolvent, its spectrum is given by

σ(A) = iµj : j ∈ Λ with Λ = Z∗ or N∗ and where (µj)j∈Λ is a sequence of real numbers.

Recalling the definition of (Ψj)j∈Λ in (1.37), we have the following Theorem (see [84]):

2.5. CONSERVATIVE LINEAR SYSTEMS 29

Theorem 2.5.2. For ω ∈ lR and ε > 0, set

Jε(ω) = m ∈ Λ such that |µm − ω| < ε.

Then system (2.18)-(2.19) is exactly observable if and only if the following equivalent asser-

tions hold:

1. There exists ε > 0 and δ > 0 such that for all n ∈ Z∗ and for all z =∑

m∈Jε(µn)

cmΨm

‖Bz‖Y ≥ δ ‖z‖X .

2. There exists ε > 0 and δ > 0 such that for all ω ∈ lR and for all z =∑

m∈Jε(ω)

cmΨm

‖Bz‖Y ≥ δ ‖z‖X .

This Theorem generalizes the kind of controllability/observability results that one may

get using Ingham’s inequality ([100]). In that case the spectrum is assumed to satisfy a gap

condition and, consequently, by taking ε > 0 small enough, the sets Jε(ω) are reduced to

contain one single eigenvalue for each ω. However, in practice, Ingham inequalities can only

applied to 1− d problems while the “wave-packet” method that this Theorem proposes has

a much wider range of applicability.

In Chapter 5, we will assume that system (2.18)-(2.19) is exactly observable and analyze

its time-discrete counterpart, using resolvent estimates of the form (2.22).

Chapter 3

The time-discrete heat equation

3.1. Introduction

In this chapter we shall study first the uniform null-controllability of the time-discrete

version of (2.7) by means of the L-R method. Then we analyze the uniform approximate

controllability by means of the uniform null control. The classical known controllability

results for (2.7) are recovered.

For this purpose, we first recall the definition of t in Chapter 1.1. For any given

K ∈ lN∗, we set t = T/K and introduce the net

t0 = 0 < t1 < · · · < tK = T

with tk = kt and k = 0, 1, · · · ,K.

The time-discrete counterpart of system (2.7) reads as follows:

yk+1 − yk

t − ∆yk+1 = uk1ω, x ∈ Ω, k = 0, 1, · · · ,K − 1

yk = 0, x ∈ ∂Ω, k = 1, · · · ,Ky0 ∈ L2(Ω) given.

(3.1)

System (3.1) is an implicit time discretization of the heat equation with control on the

subset ω ⊂ Ω. Here yk0≤k≤K stands for the state and uk0≤k≤K−1 the control. There

are many methods to discretize system (2.7). We choose the Implicit Euler schemes (3.1)

simply because it avoids the instability of the solutions in the whole space.

As the analogue of the continuous case, we introduce the following two definitions for

the discrete schemes (3.1):

Definition 3.1.1. System (3.1) is said to be approximately controllable at k = K (for any

given t > 0) if for any y0 ∈ L2(Ω), any final state yT ∈ L2(Ω) and ε > 0, there exists a

31

32 CHAPTER 3. THE TIME-DISCRETE HEAT EQUATION

control uk ∈ L2(ω)0≤k≤K−1 such that the solution ykk=0,··· ,K of (3.1) satisfies

∥∥yK − yT∥∥

L2(Ω)< ε.

Definition 3.1.2. System (3.1) is said to be null controllable at k = K (for any given

t > 0) if for any y0 ∈ L2(Ω) there exists a control uk ∈ L2(ω)0≤k≤K−1, called henceforth

a discrete null control, such that the solution ykk=0,··· ,K of (3.1) satisfies

yK(x) = 0, ∀ x ∈ Ω.

For any fixed t > 0, (3.1) is a system of controlled elliptic equations.

The rest of this chapter is organized as follows. In Section 2 we first show the lack of

controllability of system (3.1). In Section 3 we roughly analyze the L-R method in the time-

discrete level. Section 4 is devoted to show our main result, in which we prove a uniform

partial controllability of system (3.1). Consequently, an error estimate is made in Section

5 to compare the time-continuous and time-discrete controls. Approximate controllability

of system (3.1) is stated in Section 6. Section 7 is devoted to analyze some other discrete

schemes , i.e. explicit Euler method, θ-method. We also introduce the time implicit semi-

discrete fractional order parabolic system in the last Section and obtain some similar results.

3.2. Lack of controllability

In this section we will show the lack of controllability of system (3.1) and the necessity

of introducing filtering. The following result shows that, whatever Ω ⊂ lRd is, controllability

properties of system (3.1) fail for any ω, unless ω covering the whole domain,i.e. ω = Ω:

Theorem 3.2.1. Let Ω \ ω 6= ∅. For any given t > 0, system (3.1) is neither null nor

approximately controllable.

Remark 3.2.1. In fact, the Hilbert Uniqueness Method (HUM, see in [57]) provides that

the null-controllability of system (3.1) is equivalent to the observability of its adjoint system

−ϕk+1 − ϕk

t − ∆ϕk = 0, x ∈ Ω, k = 0, 1, · · · ,K − 1

ϕk = 0, x ∈ ∂Ω, k = 1, · · · ,KϕK given.

(3.2)

3.2. LACK OF CONTROLLABILITY 33

The key point of the lack of null-controllability is that the adjoint system (3.2) is not ob-

servable for ϕK ∈ L2(Ω), i.e.

supϕK∈ L2(Ω)

∥∥ϕ0∥∥2

L2(Ω)

tK−1∑

k=0

∫

ω|ϕk|2dx

= ∞ (3.3)

for any fixed t > 0, except for the trivial case ω = Ω.

Proof of Theorem 3.2.1: First, let us prove that system (3.1) is not null controllable.

We use contradiction argument. If (3.1) is null controllable, then for any y0 ∈ L2(Ω), we

can find a control uk0≤k≤K−1 such that the solution yk0≤k≤K of system (3.1) vanishes

at k = K. Multiplying the first equation in (3.1) by the solution ϕk of (3.2) and summing

up in k, using integration by parts, we get

0 =

∫

ΩyKϕKdx = t

K−1∑

k=0

∫

ωukϕkdx+

∫

ωy0ϕ0dx.

Hence∫

Ωy0ϕ0dx = −t

K−1∑

k=0

∫

ωukϕkdx, ∀ y0 ∈ L2(Ω). (3.4)

We now show that (3.4) is impossible.

Since |Ω \ ω| > 0, there exists a point x0 in Ω \ ω and consequently one can find a ball

B(x0, A) = x : |x− x0| < A ⊂ Ω \ ω,

with some positive constant A. We choose a function ψ ∈ C∞0 (B) with positive L2-norm

and let ϕ0 = ψ. From system (3.2) we compute

ϕk+1 = ϕk −t∆ϕk, k = 0, · · · ,K − 1. (3.5)

Multiplying (3.5) by ϕk and integrating in Ω, we obtain

∫

Ωϕk+1ϕkdx =

∫

Ω|ϕk|2dx+ t

∫

Ω|∇ϕk|2dx. (3.6)

By (3.6) we deduce that ϕk+1 does not vanish in Ω when ϕk has positive L2-norm. Conse-

quently the initial data ϕK ∈ L2(Ω) has positive L2-norm.


Moreover, taking into account that ϕk ∈ C∞0 (B) for any k ≥ 0 and B(x0, A) ∩ ω = ∅,

we find that the right side of (3.4) vanishes, i.e.

−tK−1∑

k=0

∫

ωukϕkdx = 0. (3.7)

Hence, by taking y0 = ϕ0 in (3.4), we deduce that ϕ0 ≡ 0, which is a contradiction.

Next, we will prove that system (3.1) is not approximately controllable.

It is easy to deduce that approximate controllability of (3.1) is equivalent to the unique

continuation of (3.2).

However, it is obvious that the adjoint system (3.2) is not observable since as shown

above, there exists a special initial data ϕK ∈ C∞0 (Ω \ ω) such that the corresponding

solution ϕkk=0,··· ,K−1 of (3.2) satisfies:

∥∥ϕk∥∥

L2(ω)= 0 for any k = 0, · · · ,K − 1;

∥∥ϕ0∥∥

L2(Ω)> 0.

This fact shows that the unique continuation of (3.2) fails. Roughly speaking, the informa-

tion of ϕK never appears in the subdomain ω. The observability of system (3.2) fails even

with initial data containing in C∞0 (Ω \ ω).

Consequently, the approximate controllability of (3.1) fails.

Remark 3.2.2. There is an equivalent assertion of the approximate controllability property

of the control system, named as “Unique Continuation” for solutions of its adjoint system

(see in [72]). It is easy to show the equivalence also holds in the time-discrete level. More

precisely, one can prove that, the approximate controllability of (3.1) is equivalent to the

unique continuation for solutions of system (3.2), which is defined by:

The solution of system (3.2) is said to be of unique continuation if and only if the

solution ϕkk=0,··· ,K satisfies:

ϕk = 0 in Ω,∀ k = 0, · · · ,K, ⇔ ϕk = 0 in ω,∀ k = 0, · · · ,K.

Another way to prove Theorem 3.2.1 is to find a counterexample such that the above asser-

tion fails. It is straightforward and we omit the details here.

3.3. HEURISTICS OF THE L-R METHOD 35

However, the counterexample in the proof of Theorem 3.2.1 disappears if we consider

ϕK has the form as a finite combination of the Fourier expansion, i.e.

ϕK =∑

µ2j≤σ

ajΦj(x), (3.8)

with a positive constant σ > 0. It is well-known that the function ϕK defined in (3.8) is

analytic in Ω and only contains finite number of zero points (see [46]). Hence ϕK no longer

belongs to C∞0 (Ω \ ω).

We will see in the next section, if ϕK has the form as in (3.8), the solutions of (3.2)

are observable. Consequently, it is possible to discuss the controllability of system (3.1) by

the duality argument. This technique is called “filtering method”, and has been success-

fully applied in the context of controllability of numerical approximate schemes for wave

equations (see [110] and the references therein).

3.3. Heuristics of the L-R method

Let us consider now the time-discrete heat equation (3.1) by means of the L-R method.

Recall that yk is the solution and uk is the corresponding control at time step k. When

uk = 0 for 0 ≤ k ≤ K − 1, i.e. no control acts on the domain ω, the solution of (3.1) at

t = T , i.e. at k = K, can be written by means of the Fourier expansion as

yK =∑

j≥1

aj(1 + µ2jt)−KΦj =

∑

j≥1

aj(1 + µ2jt)−T/tΦj. (3.9)

Assume σ is a positive constant, then yK decays at the rate (1+σt)−T/t with initial

data y0 =∑

µ2j≥σ ajΦj.

From the L-R time iteration method, we see that the key point to obtain a bounded

control is that the norm of the solution of the continuous heat equation decays exponentially

at the order of exp(−σt). This phenomena compensates the exponential increase of the norm

of control at the order of exp(C√σ). However, the solution of the time-discrete system (3.1)

decays at the order of (1+σt)−T/t. We need to know, under which condition this decay

of solutions can compensate the exponential increase of the norm of control at the order of

exp(C√σ) in the time-discrete case.

For this purpose, we show the following elementary result:


Lemma 3.3.1. Let T and C > 0 be two positive constants and 0 < t < T 2

4√

2C2. Then

function

f(σ) = eC√

σ(1 + σt)−[ Tt

] (3.10)

has the following properties:

i). f(σ) is decreasing in the interval((C

T )2, ( T2Ct)

2).

ii). f(σ) < e−δ√

σ in the interval(( 2C+δ

T−t)2, ( T

2(C+δ)t )2)

for any 0 ≤ δ ≤ C.

iii). It holds

limt→0+

f(σ)∣∣∣σ=( T

Ct)2

= 0. (3.11)

Proof: Replacing√σ by x and setting f(x2) = g(x), we have

g(x) = exp(Cx− [

T

t ] ln(1 + t x2)).

The derivative of f(x2) with respect to x reads

g′(x) = g(x)(C − [

T

t ]2tx

1 + tx2

).

g(x) is a decreasing function if and only if g′(x) is negative. Since [ Tt ] >

Tt − 1, by solving

the inequality

C − 2(T −t)x1 + t x2

< 0,

we conclude that g(x) decreases in the interval

( (T −t) −√

(T −t)2 − C2tCt ,

(T −t) +√

(T −t)2 − C2tCt

).

Moreover, taking into account that f(σ) = f(x2) = g(x) and

max0<t<( T

2C)2

(T −t) −√

(T −t)2 − C2tCt

= max0<t<( T

2C)2

C

(T −t) +√

(T −t)2 − C2t≤ C

T,

(min0<t<( T

2C)2

(T −t) +√

(T −t)2 − C2tC

) 1

t ≥T

2C

1

t ,

we conclude that f(σ) is decreasing in the interval((C

T )2, ( T2Ct)

2)

for any t ∈ (0, (TC )2).

3.3. HEURISTICS OF THE L-R METHOD 37

Now we consider the function H(x) = f(x2)eδ√

σ. Clearly, H(x) < 1 if and only if

lnH(x) = C x− [T

t ] ln(1 + t x2) + δ x < 0.

By the Taylor expansion, lnH(x) can be rewritten as:

lnH(x) = (C+δ)x−[T

t ]t x2+[

T

t ](t)2

2x4−[

T

t ](t)3x6

3(1 + ξt x2)3, for some ξ ∈ (0,t x2).

Since x > 0, we deduce that lnH(x) < 0 if

C + δ − (T −t)x+Tt

2x3 < 0. (3.12)

We claim that (3.12) is satisfied when x ∈(

2C+δT−t , (

2CTt)

1/3). This is due to the fact that

C + δ − (T −t)x+Tt

2x3 < −C +

Tt2

x3 < 0.

Similarly, it is easy to show that H(x) is decreasing for x ∈ (C+δT , T

2(C+δ)t ). Hence,

H(x) < 1 when x satisfies

2C + δ

T −t < x < max((

2C

Tt)1/3,

T

2(C + δ)t). (3.13)

Moreover, taking t ∈ (0, T 2√2(C+δ)2

), we get that the right side of (3.13) equals to T2(C+δ)t

and consequently H(x) < 1 in the interval

2C + δ

T −t < x <T

2(C + δ)t .

Hence, ii) is satisfied when t < min0<δ<CT 2

√2(C + δ)2

, i.e. t < T 2

4√

2C2.

Next, by

limt→0

f(σ)∣∣∣σ=( T

Ct)2

= limt→0

exp(T − [ T

t ]t ln(1 + T 2

C2t)

t)

= 0, (3.14)

we get (3.11).

Remark 3.3.1. The function f(σ) comes from the L-R estimate and indicates the com-

pensation between the increase of control and the decay of the solution. Lemma 3.3.1 tells

that the increase of the norm of the control can be compensated uniformly with respect to

t, provided that eigenvalues σ satisfy σ ≤ ( T2Ct)

2.


Remark 3.3.2. Now we denote the time parameter T by Tl which indicates the length of

the time interval of the l-th step iteration in the L-R time iteration method in our proof

of Theorem 3.4.1. Moreover, let T > 0 be the control time in Theorem 3.4.1, we have the

following identity between T and Tl:

Tl = 2−l−1T,

where l is a positive integer. Obviously Tl < T and we will see, by careful analysis in Section

4, function (3.10) makes sense as σ increasing as the order of (t)−r with r ∈ (0, 2), instead

of (t)−2.

3.4. Partial null-controllability

In view of the lack of controllability of system (3.1) it is natural to relax the control-

lability requirement by considering the projections of solutions over a suitable class of low

frequency Fourier components. In fact, it is by now well-known that, often, numerical ap-

proximations schemes that are stable develop instabilities when applied to controllability

problems. It is due to the presence of spurious high frequencies numerical solutions that

the control mechanisms are not able to control uniformly as the mesh-size tends to zero.

Hence, it is reasonable to cut off the high frequencies and only consider the lower part. This

filtering method has been applied successfully in the context of controllability of numerical

approximate schemes for wave equations (see [110] and the references therein).

Recalling the definition of C0,s in (1.5) and the projection operator π0,s on C0,s, the

following result shows that the projection of the solution of system (3.1) on C0,s is uniformly

null controllable, with appropriate choice of s:

Theorem 3.4.1. For any fixed T > 0 and r ∈ (0, 2), there exists a positive constant

Λ = Λ(r, T,Ω, ω) such that for all y0 ∈ L2(Ω), there exists a control uk ∈ L2(ω)k=0,··· ,K−1,

so that


π0,Λ(t)−ryK(x) = 0, ∀ x ∈ Ω; (3.15)

(2) There exists a constant C = C(r, T,Ω, ω) > 0, independent of t, such that

tK−1∑

k=0

∫

ω|uk|2dx ≤ C

∫

Ω|y0|2dx (3.16)

3.4. PARTIAL NULL-CONTROLLABILITY 39

holds for any t > 0 and y0 ∈ L2(Ω).

Some remarks are in order.

Remark 3.4.1.

1. Note that when t is fixed, system (3.1) is also null controllable for any filtering

parameter s > 0. This is due to the duality and the fact that its adjoint system is

observable after filtering, no matter what “s” is. However, from the proof of Theorem

3.4.1, we shall see that in order to keep the controllability property uniformly as t→0, one needs to choose the filtering parameters to be of the order Λ(t)−r with

Λ ∼( T

8D

)r. (3.17)

In (3.17), T is the control-time and D is a constant depending on Ω and ω. Moreover,

for any two controllers ω1 and ω2 in Ω, D(ω1) > D(ω2) whenever ω1 ⊂ ω2, and,

accordingly, Λ(ω1) < Λ(ω2). Furthermore, Λ increases as the time T increases.

2. Note that for any r ∈ (0, 2) fixed, when t tends to zero, the filtering parameter

s = Λ(t)−r tends to infinity and the filtered space Cs tends to cover the whole space

L2(Ω). We imposed the restriction r ∈ (0, 2) for technical reasons. However, it is

likely that, when r = 2, the result fails because of the lack of sufficient dissipation,

as it happens in the critical fractional order heat equation (see [67]). This is an

interesting problem.

3. The results of this Chapter concern the interior control problem. The same issues make

sense in the context of the boundary controllability. Recall that the time-continuous

heat equation is controllable for all time T > 0 and an arbitrarily support Γ0, open

nonempty subset of Γ (see, for instance, [49]). However, even in the time continuous

case, one can not derive the boundary controllability directly by means of the ana-

logue introduced here since (2.10) is false when the observation set is a subset of the

boundary. This is obvious, in particular, in the 1 − d case.

A rather standard method to derive the boundary controllability from the interior one

is based on extension-restriction argument and it is as follows. One first extends by

zero the solution to an outer neighborhood of Γ0. The arguments for interior controls

allow to control the system in the whole domain by means of a control supported in this

small added domain. The restriction of the solution to the original domain satisfies


all of the requirements and its restriction or trace to the subset of the boundary where

the control had to be supported, provides the boundary control one is looking for.

However, this argument does not work well for the present discrete problem. Indeed,

by doing this, of course one can obtain a uniform (partial) null-controllability of the

system after filtering. However, the filtering space is spanned by the eigenvectors of

the Dirichlet Laplacian in the extended domain rather than the original domain Ω.

Very likely, in this time-discrete setting, the most promising technique seems to be that

based on the use of Carleman inequalities. But so far there have not been addressed

in the time-discrete setting.

4. Note that system (3.1) is a scheme discrete in time and continuous in space. Naturally,

as a further study, one could consider the control problem for fully discrete schemes,

both on time and space variables, for instance that one replaces ∆ in (3.1) by a finite-

difference space discretization operator. As far as we know, the controllability of such a

fully discrete scheme is an open problem. The difficulty is that it is not clear whether

the space discrete version of (2.10) holds or not, which seems to be a challenging

problem. Indeed, the proof of (2.10) is based on doubling properties or Carleman

inequalities for the space-continuous elliptic equations. However, none of these tools

are developed well in the discrete settings. At this level it is very likely that one possible

method to be explored could be the time-discrete biorthogonal sequences, as a discrete

counterpart of the existing theorem for time-continuous 1−d parabolic problem ([17]).

5. Similar discrete controllability results can also be obtained in a more general set-

ting. For instance, the operator ∆y can be replaced by∑d

i,j=1(aij(x)yxi)xj , where(aij

)1≤i,j≤d

∈ C∞(Ω; lRd×d) is a symmetric and uniformly positive definite matrix.

Indeed, in this case, as shown in [52], the counterpart of (2.10) holds true. Very likely,

one may even adding a zero-order term ay (with a ∈ L∞(Ω)) to∑d

i,j=1(aij(x)yxi)xj

since the counterpart of (2.10) for the resulting elliptic operator should hold true (Re-

call that (2.10) can be proved by means of Carleman estimate, which is “robust” with

respect to bounded perturbations). However, when adding any nonzero first-order term

to∑d

i,j=1(aij(x)yxi)xj , the resulting operator is not self-adjoint any more. As for other

boundary conditions, the problem is, again, whether or not, the counterpart of (2.10)

remains to be true, which, as far as we know, is an unsolved problem.

The main technique used in the proof of the above Theorem is the L-R method, which


also plays a key role in the proof of the continuous case.

Proof of Theorem 3.4.1: We use the L-R time iteration method. The proof is split

into several steps.

Step I. Let us show the partial controllability property for system (3.1).

We consider first a partial observability of (3.2), i.e. the adjoint system of (3.1). Denote

by ajj≥1 the Fourier coefficients of ϕK , i.e.

ϕK =∑

j≥1

ajΦj.

The solution of system (3.2) is given by:

ϕk =∑

j≥1

aj(1 + µ2jt)k−KΦj.

It is easy to see that, for any ϕK ∈ C0,σ, its Fourier coefficients aj = 0 whenever µ2j ≥ σ.

For any σ > 0, we claim that there exist two positive constants Cj = Cj(Ω, ω) > 0, j = 1, 2,

independent of σ, such that

tK−1∑

k=0

∫

ω|ϕk|2dx ≥ C1Te

−C2√

σ

∫

Ω|ϕ0|2dx, ∀ ϕK ∈ C0,σ. (3.18)

In fact, using inequality (2.10) in Theorem 2.3.2 and recalling Kt = T , we have

tK−1∑

k=0

∫

ω|ϕk|2dx = t

K−1∑

k=0

∫

ω

∣∣∣∑

µ2j≤σ

aj(1 + µ2jt)k−KΦj

∣∣∣2dx.

Moreover, it is easy to show that

tK−1∑

k=0

∫

ω|ϕk|2dx ≥ t

K−1∑

k=0

C1e−C2

√σ∑

µ2j≤σ

|aj(1 + µ2jt)k−K |2

= C1e−C2

√σ∑

µ2j≤σ

a2jt

K∑

k=1

(1 + µ2jt)−2k

≥ C1Te−C2

√σ∑

µ2j≤σ

a2j(1 + µ2

jt)−2K

= C1Te−C2

√σ

∫

Ω|ϕ0|2dx.

Formula (3.18) is a partial observability of system (3.2). By means of the classical duality

argument we conclude that there exist uk0≤k≤K and two positive constants C1, C2 > 0,

such that the corresponding solution yk0≤k≤K of (3.1) satisfies

π0,σyK = 0, (3.19)


and

tK∑

k=0

∫

ω|uk|2dx ≤ 1

C1TeC2

√σ

∫

Ω|y0|2dx. (3.20)

Step II. We now construct the desired control by means of the L-R method.

For l = 1, 2, ..., we set σl = Aσl−1 with two parameters σ0 > 0 and A > 1 to be

determined later. Set T0 = 0, T2l − T2l−1 = T2l−1 − T2l−2 = 2−l−1T . More precisely, we

choose

T2l = (1 − 2−l)T, T2l−1 = (1 − 3 · 2−l−1)T, l = 1, 2, · · · .

The time interval (0, T ) is divided to a series of subintervals

I1 = (T0, T2), I2 = (T2, T4), · · · , Il = (T2l−2, T2l), · · · . (3.21)

We choose the control for system (3.1) as follows:

• In the time interval (T0, T1). This is the first half part of I1. Set K0 = [ T0t ] and

K1 = [ T1t ]. From Step I, especially recalling (3.19) and (3.20), we can find a control

ukK0≤k≤K1 (3.22)

such that the corresponding solution ykK0≤k≤K1 of (3.1) satisfies

π0,σ1yK1 = 0, (3.23)

and

tK1∑

k=K0

∫

ω|uk|2dx ≤ 1

C1(T1 − T0)eC2

√σ1

∫

Ω|y0|2dx. (3.24)

By means of the usual energy method, noting (3.24) and T1−T0 = 2−2T , it is clear that∫

Ω|yK1|2dx ≤ eD

√σ1

∫

Ω|y0|2dx, (3.25)

where the constant D > 0 is independent of t.

• In the time interval (T1, T2). This is the last half part of I1. Set K2 = [ T2t ]. We

choose the control as

uk = 0, k = K1 + 1, · · · ,K2. (3.26)

Taking (3.23) into account and T2 − T1 = 2−2T , it is easy to show that∫

Ω|yK2 |2dx ≤ (1 + σ1t)−[ 2

−2Tt

]∫

Ω|yK1|2dx. (3.27)


• In the time interval (T2l−2, T2l−1), with l = 2, 3, · · · . This is the first half part of

Il. Set K2l−2 = [T2l−2

t ] and K2l−1 = [T2l−1

t ]. Similarly, one can find a control

ukK2(l−1)<k≤K2l−1(3.28)

such that

π0,σlyK2l−1 = 0, (3.29)

and

tK2l−1∑

k=K2(l−1)

∫

ω|uk|2dx ≤ 1

C1(T2l−1 − T2(l−1))eC2

√σl

∫

Ω|yK2(l−1) |2dx. (3.30)

By means of usual energy method, noting (3.30) and T2l−1−T2(l−1) = 2−l−1T , it is clear

that ∫

Ω|yK2l−1 |2dx ≤ eD

√σl

∫

Ω|yK2(l−1) |2dx, (3.31)

with the same constant D in (3.25).

• In the time interval (T2l−1, T2l). This is the last half part of Il. Set K2l = [T2lt ].

We choose the control as

uk = 0, k = K2l−1 + 1, · · · ,K2l. (3.32)

Taking (3.29) and (3.31) into account and recalling T2l −T2l−1 = 2−l−1T , it is easy to show

that ∫

Ω|yK2l |2dx ≤ (1 + σlt)−[ 2

−l−1Tt

]∫

Ω|yK2l−1 |2dx

≤ (1 + σlt)−[ 2−l−1Tt

]eD

√σl

∫

Ω|yK2(l−1) |2dx.

(3.33)

By induction, we have

∫

Ω|yK2l |2dx ≤

l∏

s=1

(1 + σst)−[ 2−s−1Tt

]eD

√σs

∫

Ω|y0|2dx. (3.34)

Replacing l by l − 1 in (3.34) and recalling (3.30) we deduce that there exists a constant

C > 0, depending only on C1, such that

tK2l−1∑

k=K2(l−1)

∫

ω|uk|2dx ≤ CeC2

√σl

l−1∏

s=1

(1 + σst)−[ 2−s−1Tt

]eD

√σs

∫

Ω|y0|2dx

≤ CeD√

σ1

l−1∏

s=1

(1 + σst)−[ 2−s−1Tt

]eD√

σs+1

∫

Ω|y0|2dx.

(3.35)


Step III. Recalling that σl = Aσl−1 with A > 1, we may rewrite the product∏l−1

s=1(1 + σst)−[ 2−s−1Tt

]eD√

σs+1 in (3.35) as a function

R(σl) =l−1∏

s=1

exp(D√Aσs −

[2−s−1T

t]ln(1 + σst)

). (3.36)

By Lemma 3.3.1 we claim that for 0 ≤ δ ≤ D√A, there exists a positive constant C3,

depending on σ0 and A, such that

R(σl) ≤l−1∏

s=1

e−δσs ≤ C3e− δ

A−1σl (3.37)

holds under the restrictions

( 2D√A+ δ

2−s−1T −t)2

≤ σs ≤( 2−s−1T

2(D√A+ δ)t

)2. (3.38)

We now choose σ0 and A such that (3.38) holds. Letting δ = D√A. First, it is not

difficult to see that, the left inequality of (3.38) is satisfied if

t ≤ 2−s−2T, A ≥ 4 and σ0 ≥ (4

A)l−1 256D2

T 2. (3.39)

Secondly, by the second inequality in (3.38), we need

σl = Alσ0 ≤ 4−lT 2

64D2A(t)2 , (3.40)

or, equivalently,

l ≤ log4A

( T 2

64σ0D2(t)2).

Denoting by L the maximum of l, i.e.

L =[log4A

( T 2

64σ0D2(t)2)], (3.41)

one has the estimate

σL = ALσ0 ≥ Alog4A

(T2

64σ0D2(t)2

)−1σ0 = σ

log4A 40

( T

8D

)2(1−log4A 4)(t)−2(1−log4A 4).

Hence, for any r ∈ (0, 2), we choose

A = max(4,1

4e

2 ln 21−r/2 ) and σ0 ≥ 256D2

T 2. (3.42)

Consequently we have σL ≥ Λ(t)−r, with a constant Λ = Λ(r) independent of t. More

precisely,

3.5. ERROR ESTIMATES 45

For r ∈ [1, 2), we have σL ≥ Λ(t)−r, with Λ = Λ(r) = σ1− r

20

(T8D

)r.

For r ∈ (0, 1), we have σL ≥ Λ(t)−1 ≥ Λ(t)−r, with Λ = Λ(1) = σ1/20

(T8D

).

Step IV . From the analysis above, we know that each t corresponds to a filtering

parameter σL. Combining all of the control (3.22), (3.26), (3.28), (3.32) and setting

uk = 0, K2L < k ≤ T/t,

we conclude that for any r ∈ (0, 2) there exist a series of ukk=0,··· ,K−1 and a positive

constant Λ = Λ(r, T,Ω, ω) such that

π0,Λ(t)−ryK(x) = 0, ∀ x ∈ Ω. (3.43)

Moreover, combining (3.24),(3.35) and taking (3.36) into account, we conclude that

tK−1∑

k=0

∫

ω|uk|2dx ≤ C1e

D√

σ1

(1 +

L−1∑

l=1

R(σl)) ∫

Ω|y0|2dx. (3.44)

The analysis in the third step yields a constant C, independent of t, such that

tK−1∑

k=0

∫

ω|uk|2dx ≤ C

∫

Ω|y0|2dx (3.45)

for any t > 0. More precisely, the constant is given by

C = supt>0

C1eD√

σ1

(1 + C3

L−1∑

l=1

e−D

√A

A−1σl

)

≤ C1eD√

σ1

(1 + C3

∞∑

l=1

e−D

√A

A−1σl

)

= C(A,σ0, T, C1, C2),

which dependents only on r, T,Ω and ω.

3.5. Error estimates

In this section we analyze the convergence and error estimate for the discrete null con-

trols. We have the following result:

Theorem 3.5.1. For the discrete null control uk0≤k≤K−1 given in the proof of Theorem

3.4.1, it holds

UK(·, x) =

K−1∑

k=0



where u is a null control of system (2.7). Moreover, there exists a constant C > 0, inde-

pendent of t and y0, such that UK and u satisfy

∥∥UK − u∥∥

L2(ω)≤ C

√t∥∥y0∥∥

L2(Ω). (3.46)

Proof of Theorem 3.5.1: Let Tl, Il, ukk=0,··· ,K be the same as in (3.21). From

the proof of Theorem 3.4.1 we know that the following properties of the discrete control

ukk=0,··· ,K−1 hold:

For any fixed t > 0, only L steps of time iterations are given by L-R method, where

L can be deduced by (3.41);

L tends to infinity as t tends to zero;

The control is separately located in the subintervals (T2(l−1), T2l−1), with l = 1, · · · , L.

Hence, we consider the convergence of control in each subinterval (T2(l−1), T2l−1), sepa-

rately.

Step 1: First we consider the discrete control ukk=0,··· ,K1 with K1 = [ T1t ], i.e. the

control in the time subinterval (T0, T1).

• Discrete control. Let ψK1 ∈ C0,σ1 be the initial data of (3.2), i.e. the adjoint

system of (3.1). We denote the corresponding solution by ψkk=0,··· ,K1.

Given y0 ∈ L2(Ω), the control uk ∈ L2(ω)k=0,··· ,K1 is given by

uk = ψk, ∀ k = 0, · · · ,K1. (3.47)

Here ψkk=0,··· ,K1 is the solution of system (3.2) corresponding to the initial data ψK1

which minimizes the functional

Jt(ψK1) =

1

2t

K1−1∑

k=0

∫

ω|ψk|2dx+

∫

Ωy0ψ0dx (3.48)

in the space C0,σ1 . The minimizer of Jt is well defined since observability of the system

(3.2) is provided if its initial data is taken in C0,σ1 .

Taking into account that ψK1 is the minimizer, we deduce that ψkk=0,··· ,K1 satisfies

tK1−1∑

k=0

∫

ωψkψkdx+

∫

Ωy0ψ0dx = 0, ∀ ψK1 ∈ C0,σ1 . (3.49)


We define two functions ψt and ψt by

ψt(t, ·) =

K1−1∑

k=0

ψk(·)1[kt,(k+1)t)(t), ψt(t, ·) =

K1−1∑

k=0

ψk(·)1[kt,(k+1)t)(t),

respectively. Consequently, formula (3.49) can be rewritten as a time-continuous form∫ T1

0

∫

ωψtψtdxdt +

∫

Ωy0ψ0dx = 0, ∀ ψK1 ∈ C0,σ1. (3.50)

• Continuous control. Let ρT1 ∈ C0,σ1 be the initial data of (2.8), i.e. the adjoint

system of (2.7), with T replaced by T1. We denote the corresponding solution by ρ(t, x)

with t ∈ (0, T1).

Similarly, given the same initial data y(0, x) = y0 for the continuous system (2.7), the

control u(t, x) ∈ L2((0, T1) × ω) is given by

u(t, x) = ρ(t, x), (t, x) ∈ (0, T1) × ω, (3.51)

where ρ is the solution of (2.8) corresponding to the initial data ρT1 which minimizes the

functional

J(ρT1) =1

2

∫ T1

0

∫

ωρ2dxdt +

∫

Ωy0ρ(0)dx (3.52)

in the space C0,σ1.

Taking into account that ρT1 is the minimizer of J(ρT1), we conclude that ρ satisfies∫ T1

0

∫

ωρρdxdt+

∫

Ωy0ρ(0)dx = 0, ∀ ρT1 ∈ C0,σ1 . (3.53)

Now we compute the difference between ψt and ρ.

Taking ψK1 = ρT1 ∈ C0,σ1 , combining (3.50) and (3.53) we get

∫ T1

0

∫

ω(ψt − ρ)ρdxdt =

∫

Ωy0(ρ(0) − ψ0)dx+

∫ T1

0

∫

ωψt(ρ− ψt)dxdt. (3.54)

Hence,

∣∣∣∫ T1

0

∫

ω(ρ− ψt)ρdxdt

∣∣∣

≤∥∥y0∥∥

L2(Ω)

∥∥ρ(0) − ψ0∥∥

L2(Ω)+∥∥∥ψt

∥∥∥L2((0,T1)×ω)

∥∥ρ− ψt∥∥

L2((0,T1)×ω).

(3.55)

Assuming ψK1 = ρT1 =∑

µ2j≤σ1

ajΦj, we get

ρ =∑

µ2j≤σ1

aje−µ2

j (T1−t)Φj , ψk =∑

µ2j≤σ1

aj(1 + µ2jt)−(K1−k)Φj. (3.56)


The square of∥∥ρ(0) − ψ0

∥∥L2(Ω)

reads

∥∥ρ(0) − ψ0∥∥2

L2(Ω)=

∫

Ω

( ∑

µ2j≤σ1

aj

[e−µ2

j T1 − (1 + µ2jt)−K1

]Φj

)2dx

=∑

µ2j≤σ1

a2j

[e−µ2

jT1 − (1 + µ2jt)−K1

]2

=∑

µ2j≤σ1

a2je

−2µ2j T1

[1 − exp(µ2

jT1 −K1 ln(1 + µ2jt))

]2

≤(T1

2σ2

1t+O((t)2))2 ∑

µ2j≤σ1

a2je

−2µ2jT1

≤ C(σ2

1t)2 ∑

µ2j≤σ1

a2je

−2µ2j T1 = C

(σ2

1t)2

‖ρ(0)‖2L2(Ω) .

(3.57)

Similarly, we estimate

∥∥ρ− ψt∥∥2

L2((0,T1)×ω)=

K1−1∑

k=0

∫ (k+1)t

kt

∫

ω

( ∑

µj≤σ1

aj(eµ2

j (t−T1) − (1 + µ2jt)k−K1)Φj

)2dx

≤ C(σ2

1t)2∫ T1

0

∫

Ω

( ∑

µ2j≤σ1

a2je

−2µ2j (T1−t)

)dxdt

= C(σ2

1t)2

‖ρ‖2L2((0,T1)×Ω) .

(3.58)

Substituting (3.57) and (3.58) into (3.55) and using the known observability estimate

for ρ, we conclude that for any ψK1 = ρT1 ∈ C0,σ1 it holds

∣∣∣∫ T1

0

∫

ω(ψt − ρ)ρdxdt

∣∣∣

≤ Cσ21t

(‖ρ(0)‖L2(Ω) ‖y0‖L2(Ω) +

∥∥∥ψt∥∥∥

L2((0,T1)×ω)‖ρ‖L2((0,T1)×Ω)

)

≤ Cσ21t

(‖ρ‖L2((0,T1)×ω) ‖y0‖L2(Ω) +

∥∥∥ψt∥∥∥

L2((0,T1)×ω)‖ρ‖L2((0,T1)×Ω)

).

(3.59)

On the other hand, (3.54) can be also written as

∫ T1

0

∫

ω(ψt − ρ)ψtdxdt =

∫

Ωy0(ρ(0) − ψ0)dx+

∫ T1

0

∫

ωρ(ρ− ψt)dxdt. (3.60)

With the same procedure as before we have

∀ ψK1 = ρT1 ∈ C0,σ1 ,∣∣∣∫ T1

0

∫

ω(ψt − ρ)ψtdxdt

∣∣∣

≤ Cσ21t

(‖ρ‖L2((0,T1)×ω) ‖y0‖L2(Ω) + ‖ρ‖L2((0,T1)×ω) ‖ρ‖L2((0,T1)×Ω)

).

(3.61)


Now we choose ρT1 = ρT1 in (3.59) and ψK1 = ψK1 in (3.61), then adding (3.59) and

(3.61), we get

∣∣∣∫ T1

0

∫


∣∣∣+∣∣∣∫ T1

0

∫


∣∣∣

≤ Cσ21t

(‖ρ‖L2((0,T1)×ω) ‖y0‖L2(Ω) +

∥∥∥ψt∥∥∥

L2((0,T1)×ω)‖ρ‖L2((0,T1)×Ω) +

+ ‖ρ‖L2((0,T1)×ω) ‖ρ‖L2((0,T1)×Ω)

).

(3.62)

In view of the Theorem 2.3.2 and using the observability (2.10), we directly get that

‖ρ‖L2((0,T1)×Ω) ≤ C1eC2

√σ1 ‖ρ‖L2((0,T1)×ω) , (3.63)

where the two positive constants are independent of t and T1.

Since ρ and ψk are controls, we deduce from Theorem 3.4.1 that there exists a constant

C, independent of t, such that

‖ρ‖L2((0,T1)×ω) ≤ C ‖y0‖L2(Ω) ,∥∥∥ψt

∥∥∥L2((0,T1)×ω)

≤ C ‖y0‖L2(Ω) . (3.64)

Combining (3.62), (3.63) and (3.64) we conclude that

∣∣∣∫ T1

0

∫


∣∣∣+∣∣∣∫ T1

0

∫


∣∣∣

≤ Ct σ21e

C2√

σ1 ‖y0‖2L2(Ω) .

(3.65)

Consequently, ∫ T1

0

∫

ω(ψt − ρ)2dxdt ≤ C t σ2

1eC2

√σ1 ‖y0‖2

L2(Ω) . (3.66)

Step l, with l = 2, · · · , : We redo the same proceeding as in Step 1. Recalling

inequality (3.34) we get the following estimate for the initial data of the l step:

∥∥yK2(l−1)∥∥2

L2(Ω)≤ R(σl+1)

∥∥y0∥∥2

L2(Ω).

Hence, by carefully computation we obtain that

∣∣∣∫ T2l−1

T2(l−1)

∫

ω

( K2l−1∑

k=K2(l−1)

ψk1[tk ,tk+1)(t) − ρ)2dxdt

∣∣∣

≤ C t σ2l e

C2√

σl∥∥yK2(l−1)

∥∥2

L2(Ω)

≤ C t σ2l e

C2√

σlR(σl+1)∥∥y0∥∥2

L2(Ω).

(3.67)


Conclusion: Taking into account that uk = ψk for all k = 0, · · · ,K − 1 and u = Ψ,

combining inequalities (3.66) and (3.67) (applying for all 2 ≤ l ≤ L), recalling the estimation

of R(σl) in (3.37) we attain

∫ T2l−1

0

∫

ω(UK − u)2dxdt =

L∑

l=1

∣∣∣∫ T2l−1

T2(l−1)

∫

ω

( K2l−1∑

k=K2(l−1)

ψk1[tk ,tk+1)(t) − Ψ)2dxdt

∣∣∣

≤ tL∑

l=1

C σ2l e

C2√

σlR(σl)∥∥y0∥∥2

L2(Ω)≤ Ct

∥∥y0∥∥2

L2(Ω).

(3.68)

Furthermore, since UK = 0 for t ∈ (T2L−1, T ) and u(t) = 0 for t ∈ (T2L−1, T2L), we estimate

∫ T

T2L−1

∫

ω(UK − u)2dxdt =

∫ T

T2L

∫

ωu2dxdt ≤ Ce−CσL

∥∥y0∥∥2

L2 ≤ Ce−Λ(t)−r ∥∥y0∥∥2

L2(Ω).

(3.69)

Combining (3.68) and (3.69), we arrive at

∫ T

0

∫

ω(UK − u)2dxdt ≤ Ct

∥∥y0∥∥2

L2(Ω), (3.70)

which equals to (3.46).

On the other hand, from (3.45) we have

limt→0

∥∥UK∥∥2

L2((0,T )×ω)= lim

t→0t

K−1∑

k=0

∫

ω|uk|2dx < C

∫

Ω|y0|2dx <∞. (3.71)

Taking (3.70) and (3.71) into account, we arrive at

limt→0

UK = limt→0

K−1∑

k=0

uk(x)1[tk ,tk+1)(t) −→ u strongly in L2((0, T ) × ω),

where u is a control for the continuous heat equation (2.7) with initial data y0 ∈ L2(Ω).

In fact, from the construction of the control we have indicated that, the control u satisfies

u =

ul, t ∈ the first half part of Il, l = 1, 2, · · · ,

0, t ∈ the last half part of Il, l = 1, 2, · · · ,(3.72)

with

ul(t, x) = limt→0

T2l−1/t∑

k=T2l−2/t

uk(x)1[tk ,tk+1)(t).

3.6. APPROXIMATE CONTROLLABILITY 51

3.6. Approximate controllability

In previous sections, we have proved that system (3.1) is uniformly null controllable with

appropriate filtering parameter s. More precisely, after filtering the final target yK , i.e. if

we only consider the projection of yK on the filtered space C0,Λ(t)−r , the null-controllability

holds uniformly with respect to t.In this section, we further discuss the approximate controllability of system (3.1). With-

out loss of generality, we assume that y0 = 0. We have the following approximate control-

lability with uniform bounded control:

Theorem 3.6.1. Let T, r and Λ be given by Theorem 3.4.1. Then for any y0 = 0, a final

state y1 ∈ L2(Ω) and ε > 0, there exists a control ukk=0,··· ,K−1 such that yK satisfies:

∥∥π0,Λ(t)−r (yK − y1)∥∥

L2(Ω)≤ ε, ∀ t > 0. (3.73)

Moreover, for

UK =

K−1∑

k=0

uk1[tk ,tk+1)(t), (3.74)

there exists a constant C = C(r, T,Ω, ω) > 0, independent of t, such that

∥∥UK∥∥

L2((0,T )×ω)≤ C exp

(‖∆y1‖L2(Ω)

ε

)‖y1‖L2(Ω) (3.75)

holds for any t > 0 and any y1 ∈ H2(Ω) ∩H10 (Ω).

Remark 3.6.1. Note that when t is fixed, system (3.1) is approximate controllable for

any filtering parameter s > 0. This is due to the fact that it is a finite dimensional control

problem no matter what “s” is. Also when t → 0, the filtering parameter s = Λ(t)−r

tends to infinity and a continuous approximate control is obtained as the limit of the discrete

control. Moreover, the cost of the control is uniformly bounded by ‖∆y1‖L2(Ω) and ε as the

form in (3.75), which is similar to the continuous one (see the continuous result in [18]).

As a direct consequence of Theorem 3.4.1, we have the following uniform observability

of system (3.2) with ϕK ∈ C0,Λ(t)−r :

Corollary 3.6.1. Let ϕkk=0,··· ,K−1 be the solution of (3.2) with initial data ϕK . Then

for any fixed T > 0 and r ∈ (0, 2), there exists two positive constants Λ = Λ(r, T,Ω, ω) and

C = C(r, T,Ω, ω) such that

∥∥ϕ0∥∥2

L2(Ω)≤ Ct

K−1∑

k=0

∫

ω|ϕk|2dx (3.76)


holds for any t > 0 and ϕK ∈ C0,Λ(t)−r .

Proof: It is a direct consequence of Theorem 3.4.1 by means of the HUM method, we

sketch the details.

Now we give a proof of Theorem 3.6.1:

Proof of Theorem 3.6.1: In the sequel we denote Λ(t)−r by s.

For given y1 ∈ L2(Ω) and ε > 0, we define a functional Jε : C0,s −→ lR by

Jε(ϕK) =

1

2t

K−1∑

k=0

∫

ω|ϕk|2dx+ ε

∥∥ϕK∥∥

L2(Ω)−∫

Ωy1ϕ

Kdx, (3.77)

where ϕkk=0,··· ,K−1 is the solution of the adjoint system (3.2) with initial data ϕK .

First, we prove the existence of the control:

• Existence of the control. The following Lemmas 3.6.1−3.6.2 ensure that the

minimum of Jε gives a control for our approximate controllability problem:

Lemma 3.6.1. If ϕK is a minimizer of Jε in C0,s and ϕkk=0,··· ,K−1 is the solution of the

adjoint system (3.2) with initial data ϕK , then uk = ϕk|ωk=0,··· ,K−1 is a control such that

(3.73) holds.

Proof: Suppose that Jε attains its minimum value at ϕK ∈ C0,s. Then for any

ψK ∈ C0,s and h ∈ lR we have Jε(ϕK) ≤ Jε(ϕ

K + hψK). On the other hand,

Jε(ϕK + hψK)

=1

2t

K−1∑

k=0

∫

ω|ϕk + hψk|2dx+ ε

∥∥ϕK + hψK∥∥

L2(Ω)−∫

Ωy1(ϕK + hψK)dx

=1

2t

K−1∑

k=0

∫

ω|ϕk|2dx+

(t)22

tK−1∑

k=0

∫

ω|ψk|2dx+ ht

K−1∑

k=0

∫

ωϕkψkdx+

+ε∥∥ϕK + hψK

∥∥L2(Ω)

−∫

Ωy1(ϕ

K + hψK)dx.

(3.78)

Thus,

0 ≤ ε[ ∥∥ϕK + hψK

∥∥L2(Ω)

−∥∥ϕK

∥∥L2(Ω)

]+

(t)22

tK−1∑

k=0

∫

ω|ψk|2dx

+h[t

K−1∑

k=0

∫

ωϕkψkdx−

∫

Ωy1ψ

Kdx].

(3.79)


Since∥∥ϕK + hψK

∥∥L2(Ω)

−∥∥ϕK

∥∥L2(Ω)

≤ h∥∥ψK

∥∥L2(Ω)

,

we obtain that

0 ≤ ε|h|∥∥ψK

∥∥L2(Ω)

+(t)2

2t

K−1∑

k=0

∫

ω|ψk|2dx+ ht

K−1∑

k=0

∫

ωϕkψkdx− h

∫

ΩψKy1dx

holds for all h ∈ lR and ψK ∈ C0,s.

Dividing by h > 0 and by passing to the limit h→ 0 we get

0 ≤ ε∥∥ψK

∥∥L2(Ω)

+ tK−1∑

k=0

∫

ωϕkψkdx−

∫

Ωy1ψ

Kdx. (3.80)

The same calculations with h < 0 gives that

∣∣∣tK−1∑

k=0

∫

ωϕkψkdx−

∫

Ωy1ψ

K1dx∣∣∣ ≤ ε

∥∥ψK∥∥

L2(Ω), ∀ ψK ∈ C0,s. (3.81)

On the other hand, if we take the control uk = ϕk in (3.1), by multiplying in (3.1) by

ψk and adding from k = 0 to K − 1 we get that

tK−1∑

k=0

∫

ωϕkψkdx =

∫

ΩyKψKdx. (3.82)

Combining (3.81) and (3.82), it follows that

∣∣∣∫

Ω(yK − y1)ψ

Kdx∣∣∣ ≤ ε

∥∥ψK∥∥

L2(Ω), ∀ ψK ∈ C0,s,

which is equivalent to∥∥π0,s(y

K − y1)∥∥

L2(Ω)≤ ε.

The proof of the Lemma is now complete.

Now we show that J attains its minimum in C0,s.

Lemma 3.6.2. There exists a ϕK ∈ C0,s such that

Jε(ϕK) = minϕK∈C0,s

Jε(ϕK). (3.83)

Proof: It is easy to see that Jε is convex and continuous in continuous in C0,s. The

existence of a minimum of Jε is coercive, i.e.

Jε(ϕK) → ∞ when

∥∥ϕK∥∥C0,s

→ ∞. (3.84)


In fact we shall prove that

lim‖ϕK‖C0,s

→∞

Jε(ϕK)

‖ϕK‖C0,s

≥ ε. (3.85)

Obviously, (3.85) implies (3.84) and the proof of the Lemma is complete.

In order to prove (3.85) let (ϕKj ) ∈ C0,s be a sequence of initial data for the adjoint

system with∥∥∥ϕK

j

∥∥∥C0,s

→ ∞ as j → ∞. We normalize them

ϕKj = ϕK

j /∥∥ϕK

j

∥∥C0,s

,

so that∥∥∥ϕK

j

∥∥∥C0,s

= 1.

On the other hand, let ϕkj be the solutions of the (3.2) with initial data ϕK

j . Then

Jε(ϕKj )∥∥∥ϕK

j

∥∥∥C0,s

=1

2

∥∥ϕKj

∥∥Cst

K−1∑

k=0

∫

ω|ϕk

j |2dx+ ε−∫

Ωy1ϕ

Kj dx.

The following two cases may occur:

1) limj→∞

tK−1∑

k=0

∫

ω|ϕk

j |2dx > 0. In this case we obtain immediately that

Jε(ϕKj )∥∥∥ϕK

j

∥∥∥C0,s

→ ∞, as j → ∞.

2) limj→∞

tK−1∑

k=0

∫

ω|ϕk

j |2dx = 0. In this case since ϕKj is bounded in C0,s, by extracting a

subsequence we can guarantee that ϕKj ψK

0 weakly in C0,s and ϕkj ψk

0 weakly in

C0,s, where ψk0K−1

k=0 is the solution of (3.2) with initial data ψK0 . Moreover, by lower

semi-continuity,

tK−1∑

k=0

∫

ω|ψk

0 |2dx ≤ limj→∞

tK−1∑

k=0

∫

ω|ϕk

j |2dx = 0

and therefore ψk0 = 0 in ω for any k = 0, · · · ,K − 1.

On the other hand, by (3.76) it is obvious that the unique continuation of (3.2) holds

for any ψK ∈ C0,s. Hence ψk ≡ 0 in Ω for any k = 0, · · · ,K and consequently ψK0 = 0.

Therefore, ϕKj 0 weakly in C0,s and consequently

∫Ω y1ϕ

Kj dx vanishes as well.

HenceJε(ϕ

Kj )∥∥∥ϕK

j

∥∥∥C0,s

≥ limj→∞

(ε−∫

Ωy1ϕ

Kj dx) = ε,

and (3.85) follows.


Now we estimate the bound of the control ukk=0,··· ,K−1:

• Uniformly bounded control. It is an analogue of the continuous case (see, Section

6 of [18]). We assume that

z1 =

m∑

j=1

bjΦj. (3.86)

For each α > 0, let us consider the functional Jz1t,α, given by

Jz1t,α(ϕK) =

1

2

K−1∑

k=0

∫

ω|ϕk|2dx+ α

∥∥ϕK∥∥

L2(Ω)−∫

ΩϕK z1dx (3.87)

for all ϕK ∈ C0,s. Let ϕKα be the unique minimizer of Jz1

t,α in C0,s. Then since ukα =

ϕkαk=0,··· ,K−1, and ϕk

αk=0,··· ,K−1 is the solution of (3.2) and ϕK = ϕKα , we get that the

associate solution yKα of (3.1) satisfies

∥∥π0,s(yKα − z1)

∥∥L2(Ω)

≤ α, ∀ t > 0 (3.88)

Since Jz1t,α attains its minimum at ϕK

α , we have

1

2t

K−1∑

k=0

∫

ω|ϕk

α|2dx+ α∥∥ϕK

α

∥∥L2(Ω)

≤∫

ΩϕK

α z1dx.

Assuming ϕKα =

∑µ2

j≤s ajΦj and using (3.86) and (3.76), we find

tK−1∑

k=0

∫

ω|ϕk

α|2dx ≤ 2

min(s,m)∑

j=1

ajbj

≤ 2( ∑

µ2j≤s

(1 + µ2jt)−2K a2

j

)1/2( m∑

j=1

(1 + µ2jt)2Kb2j

)1/2

≤ C(t

K−1∑

k=0

∫

ω|ϕk

α|2dx)1/2

(1 + µmt)K ‖z1‖L2(Ω)

holds for any t > 0.

Hence, recalling that T = Kt, we have

(t

K−1∑

k=0

∫

ω|uk

α|2dx)1/2

=(t

K−1∑

k=0

∫

ω|ϕk

α|2dx)1/2

≤ C(1 + µmt)K ‖z1‖L2(Ω) ≤ CeµmT ‖z1‖L2(Ω) .

(3.89)

For fixed t, taking (for instance) α = 1/n for each n ≥ 1 and letting n → ∞, we can

obtain a bounded sequence of controls ukn = ϕk1ωk=0,··· ,K−1 such that the corresponding

states yKn satisfy

∥∥π0,s(yKn − z1)

∥∥L2(Ω)

≤ 1/n. (3.90)


Let ukk=0,··· ,K−1 be the weak limit in C0,s of a subsequence of uknk=0,··· ,K−1. The

corresponding solution of (3.1) is such that π0,syK = πsz1. Furthermore the function UK ,

defined in (3.74) is bounded in L2((0, T ) × ω) as in (3.89). Thus we have proved that for

any t, there exists a control UK , such that the projection of the solution of (3.1) can be

controlled exactly with cost

∥∥UK∥∥

L2((0,T )×ω)≤ CeµmT ‖z1‖L2(Ω) . (3.91)

Now assume that y1 ∈ H2(Ω) ∩H10 (Ω) and ε > 0 are given. Let us put

y1 =∑

j≥1

bjΦj, with∑

j≥1

µ4jb

2j <∞.

Let us introduce

y1,ε =

m(ε)∑

j=1

bjΦj, (3.92)

where m(ε) is such that∑

j≥m(ε)+1 b2j ≤ ε2. It is clear that

‖y1 − y1,ε‖L2(Ω) ≤ ε.

From (3.91), written for z1 = y1,ε as the form of (3.92), we obtain that there exists a control

UK such that

∥∥π0,s(yK − y1)

∥∥L2(Ω)

≤∥∥π0,s(y

K − y1,ε)∥∥

L2(Ω)+ ‖π0,s(y1,ε − y1)‖L2(Ω) ≤ ε

and, moreover, we have the following estimate

∥∥UK∥∥

L2((0,T )×ω)≤ Ceµm(ε)T ‖y1,ε‖L2(Ω) ≤ Ceµm(ε)T ‖y1‖L2(Ω) (3.93)

for any t > 0.

Notice that (3.93) must hold whenever m(ε) is such that∑

j≥m(ε)+1 b2j ≤ ε2. We are

now going to make a particular choice of m(ε) which leads to (3.75).

First, we claim that the unique case of interest is when

‖∆y1‖L2(Ω)

µ1> ε (3.94)

(recall that µ1 is the first eigenvalue of −∆ in H10 (Ω)). Otherwise ‖y1‖L2(Ω) ≤ ε and the

control uk = 0k=0,··· ,K−1 is such that the solution of (3.1) with y0 = 0 satisfies

∥∥π0,s(yK − y1)

∥∥L2(Ω)

= ‖π0,sy1‖L2(Ω) ≤ ε.

3.7. OTHER DISCRETE SCHEMES 57

Thus, the control can be zero when (3.94) is violated.

Let m(ε) be the first integer m satisfying

‖∆y1‖L2(Ω)

µm+1≤ ε

Because of (3.94), this is well defined. For this choice of m(ε), we have

∑

j≥m(ε)+1

b2j ≤ 1

µ2m(ε)+1

∑

j≥1

µ4jb

2j ≤

‖∆y1‖2L2(Ω)

µ2m(ε)+1

and, consequently, (3.93) has to be satisfied. We also have

µm(ε) ≤‖∆y1‖L2(Ω)

µm(ε)+1. (3.95)

From (3.93) and (3.95), we obtain (3.75) for any t > 0. This finish the proof of Theorem

3.6.1.

Remark 3.6.2. Note that the filtering parameter s = Λ(t)−r tends to infinity as t tends

to zero. Consequently C0,s tends to cover the whole space L2(Ω). Hence, it is easy to derive

the approximate controllability of the time continuous system (2.7), as a limit of the discrete

system.

3.7. Other discrete schemes

In this section, we consider the null-controllability of another two time-discrete schemes.

First we address the null-controllability of the Euler Explicit schemes, then we discuss the

null-controllability of the θ-method schemes.

3.7.1. The explicit Euler scheme

We state the explicit Euler time discretization of the system (2.7) as follows:

yk+1 − yk

t − ∆yk = uk1ω, x ∈ Ω, k = 0, 1, · · · ,K − 1

yk = 0, x ∈ ∂Ω, k = 1, · · · ,Ky0 ∈ L2(Ω) given.

(3.96)

Using the similar method as in the proof of Theorem 3.2.1, we claim that system (3.96) is

not null controllable, even not approximate controllable, except for the trivial case Ω = ω:


Theorem 3.7.1. Let Ω \ ω 6= ∅. Then system (3.96) is neither null controllable nor

approximate controllable for any given t > 0.

Proof: The adjoint system of (3.96) is:

−ϕk+1 − ϕk

t − ∆ϕk+1 = 0, x ∈ Ω, k = 0, 1, · · · ,K − 1

ϕk = 0, x ∈ ∂Ω, k = 0, 1, · · · ,KϕK(x) ∈ L2(Ω) given, x ∈ Ω.

(3.97)

We use the contradiction argument. If (3.96) is null controllable, then for any y0 ∈ L2(Ω)

we can find a control uk0≤k≤K−1 such that the solution yk0≤k≤K of system (3.96)

vanishes at k = K. Multiplying the first equation in (3.96) by the solution ϕk of (3.97) and

summing up in k, using integration by parts, we get

0 =

∫

ΩyKϕKdx = t

K−1∑

k=0

∫

ωukϕkdx+

∫

ωy0ϕ0dx.

Hence ∫

Ωy0ϕ0dx = −t

K−1∑

k=0

∫

ωukϕkdx, ∀ y0 ∈ L2(Ω). (3.98)

Since Ω ∩ ω∅, there exists a point x0 in Ω \ ω and consequently one can find a ball

B(x0, A) = x : |x− x0| < A ⊂ Ω \ ω,

with some positive constant A. We choose a non-trivial function ψ ∈ C∞0 (B) and let

ϕK = ψ. From system (3.97) we compute

ϕk = ϕk+1 + t∆ϕk+1, k = 0, · · · ,K − 1. (3.99)

By induction we have

ϕ0 = (I + t∆)KϕK .

Moreover, taking into account that ϕk ∈ C∞0 (B) for any k ≥ 0 and B ∩ ω = ∅, we find

that the right side of (3.98) vanishes, i.e.

−tK−1∑

k=0

∫

ωukϕkdx = 0. (3.100)

Hence, by taking y0 = ϕ0 in (3.98), we conclude that ϕ0 ≡ 0. Consequently, we have

ϕK ≡ 0 if ϕ0 ≡ 0 in Ω, which is a contradiction.


On the other hand, it is easy to prove that the unique continuation of the system (3.97)

fails. Due to the fact of the equivalence between approximate controllability of control

system and unique continuation of its adjoint system, system (3.96) is not approximately

controllable.

Let y0 =∑

j≥1 a0jΦj, u

k =∑

j≥1 bkj Φj. The solution of the system (3.96) is given by:

yk+1(x) =∑

j≥1

ak+1j Φj(x); ak+1

j = a0j (1 − µ2

jt)k+1 + 1ωtk∑

s=0

(1 − µ2jt)sbk−s

j ,

for any k = 0, 1, · · · ,K − 1.

To guarantee the stability of the scheme, we need the restriction of the eigenvalues as

µ2j ≤ (t)−1. Under this new restriction, we redo the L-R time iteration as the same process

as in the proof of Theorem 3.4.1, and attain a similar result. The only difference is that the

range of parameter r is replaced by (0, 1), instead of the range (0, 2) in the implicit Euler

case.

Proposition 3.7.1. Let ykk=0,··· ,K be the solution of system (3.96). Then Theorem 3.4.1

is true, by replacing s = Λ(t)−r with r ∈ (0, 1).

Proof: The proof is an analogue of Section 5, under an extra restriction µ2j ≤ (t)−1.

3.7.2. The θ-method

Given θ ∈ (0, 1), we discretize system (2.7) with the θ-method as follows:

yk+1 − yk

t − ∆(θyk + (1 − θ)yk+1)

)= uk1ω, x ∈ Ω, k = 0, 1, · · · ,K − 1

yk = 0, x ∈ ∂Ω, k = 1, · · · ,Ky0 ∈ L2(Ω) given.

(3.101)

The corresponding adjoint system reads:

−ϕk+1 − ϕk

t − ∆(θϕk+1 + (1 − θ)ϕk

)= 0, x ∈ Ω, k = 0, 1, · · · ,K − 1

ϕk = 0, x ∈ ∂Ω, k = 0, 1, · · · ,KϕK(x) ∈ L2(Ω) given, x ∈ Ω.

(3.102)

The solution of the system (3.102) is given by:

ϕk(x) =∑

j≥1

aj

( 1 − θµ2jt

1 + (1 − θ)µ2jt

)K−kΦj(x), ∀x ∈ Ω. (3.103)

We have the following lemma:


Lemma 3.7.1. Let T and C > 0 be two positive constants and 0 < t < min( θT 2

4C2 ,2Tθ2C

).

Then function

f(σ) = eC√

σ( 1 − θσt

1 + (1 − θ)σt)−[ T

t]

(3.104)


i). f(σ) is decreasing in the interval((C

T )2, 1θt

).

ii). f(σ) < e−δ√

σ in the interval(( 2C+δ

T−t)2, 1

θt

)for any 0 ≤ δ ≤ C.

iii). It holds

limt→0

f(σ)∣∣∣σ= 1

θt

= 0. (3.105)


g(x) = exp(Cx− [

T

t ] ln(1 + (1 − θ)tx2

1 − θtx2

)).

The derivative of g(x) with respect to x reads

g′(x) = g(x)(C − [

T

t ]2t x

(1 + (1 − θ)tx2)(1 − θtx2)

) = g(x)ρ(x).

To prove the result we need to justify that ρ(x) is negative in the interval I. It is necessary

that x2 ≤ 1θt if ρ(x) is negative. Furthermore, we compute

ρ(x) = C − [T

t ]2t x

(1 + (1 − θ)tx2)(1 − θtx2)< C − 2Tx

1 + tx2.

Obviously, by Lemma 3.3.1, ρ(x) is negative in the interval x ∈ (CT ,

T2Ct). Hence ρ(x) is

negative in the interval I = (CT , ζ) with ζ = min( 1√

θt, T

2Ct). Taking into account that

t < θT 2

4C2 , we know that ζ = 1√θt

and consequently, f(σ) is decreasing in the interval((C

T )2, 1θt

).

Now we consider the function H(x) = f(x2)eδ x. H(x) < 1 if and only if

lnH(x) = C x− [T

t ] ln(1 + (1 − θ)tx2

1 − θtx2

)+ δ x < 0.


lnH(x) = (C + ε)x−[ Tt ]t x2

1 − θtx2+

[ Tt ](t)2x4

2(1 − θtx2)2−

[ Tt ](t)3 x6

3(1 − θtx2)3(1 + ξ t x2

1−θtx2

)3


for some ξ ∈ (0, t x2

1−θtx2 ). Since x > 0, we have lnH(x) < 0 if

C + ε− (T −t)x1 − θtx2

+Ttx3

2(1 − θtx2)2< 0. (3.106)

We claim that (3.106) is satisfied when

x ∈( 2C + ε

T −t ,( 2C

4Cθt+ Tt)1/3)

.

This due to the fact that

C + ε− (T −t)x1 − θtx2

+Ttx3

2(1 − θtx2)2< −C +

Ttx3

2(1 − θtx2)2

<4Cθt x2 − 2C + Tt x3

2(1 − θtx2)2<

4Cθt x3 − 2C + Tt x3

2(1 − θtx2)2< 0.

(3.107)

In (3.107) we use x > 2C+εT−t in the first inequality and x3 < 2C

4Cθt+Tt in the last one.

Similarly, it is easy to show that H(x) is decreasing for x ∈ (C+δT , 1√

θt). Hence H(x) < 1

when x satisfies

2C + ε

T −t < x < max( 1√

θt ,( 2C

4Cθt+ Tt)1/3)

(3.108)

Moreover, taking into account that

1√θt >

( 2C

4Cθt+ Tt)1/3

for t < 2Tθ2C

we finish the proof of ii).

Next, by

limt→0

f(σ)∣∣∣σ=ζ

= limt→0

f(ζ) = f(1

θt) = 0

we get (3.105).

With Lemma 3.7.1, we obtain the similar conclusion as in Theorem 3.4.1:

Proposition 3.7.2. Let ykk=0,··· ,K be the solution of system (3.101). Then Theorem

3.4.1 is true, by replacing s = Λ(t)−r with r ∈ (0, 1) and Λ = Λ(r, θ, T,Ω, ω) > 0.

Proof: The proof is an analogue of Section 5. The only difference is that (t)2 is

replaced by t in the inequality (3.40). Consequently, we have r ∈ (0, 1).


3.8. Time-discrete fractional order parabolic equations

In this section we discuss the controllability of the time semi-discrete fractional order

parabolic equation with α > 1/2:

yk+1 − yk

t + (−∆)αyk+1 = 1ωuk, k = 0, · · · ,K − 1, x ∈ Ω

yk = 0, k = 0, · · · ,K − 1, x ∈ ∂Ω

y0 ∈ L2(Ω).

(3.109)

Equation (3.109) is the time implicit Euler semi-discretization of the continuous con-

trolled fractional order parabolic equation

yt + (−∆)αy = 1ωu, t ∈ (0, T ), x ∈ Ω

yk = 0, t ∈ (0, T ), x ∈ ∂Ω

y(0, x) = y0 ∈ L2(Ω), x ∈ Ω.

(3.110)

The controllability of equation (3.110) has been solved in [16] for the case α > 1/2 in one

space-dimension. Moreover, Micu and Zuazua (see in [67]) proved that α = 1/2 is sharp,

i.e. equation (3.110) is not controllable for α ≤ 1/2.

Therefore, in this section, we are interested in the case α > 1/2. We will prove that, the

projection of the solution of equation (3.109) is null controllable (uniformly) in any given

C0,s(recall the definition of C0,s in (1.5)):

Theorem 3.8.1. Let α > 1/2. For any fixed T > 0 and r ∈ (0, 2), there exists a positive

constant Λ = Λ(r, α, T,Ω, ω) such that for all y0 ∈ L2(Ω), there exists a control uk ∈L2(ω)k=0,··· ,K−1, so that


π0,Λ(t)−ryK(x) = 0, ∀ x ∈ Ω; (3.111)

(2) There exists a constant C = C(r, α, T,Ω, ω) > 0, independent of t, such that

tK−1∑

k=0

∫

ω|uk|2dx ≤ C

∫

Ω|y0|2dx (3.112)

holds for any t > 0 and y0 ∈ L2(Ω);

(3) The control ukk=0≤k≤K−1 of system (3.109) may be built such that

UK(·, x) =

K−1∑

k=0


where u is a null control of the corresponding continuous-time heat equation (3.110).

3.8. TIME-DISCRETE FRACTIONAL ORDER PARABOLIC EQUATIONS 63

Remark 3.8.1. Theorem 3.8.1 is the same as the Theorem 3.4.1, except for that the filtering

constant Λ is replaced by

Λ =(αT

8D

)r.

Hence, we can control more frequencies when α increases.

Remark 3.8.2. Note that when α = 1/2, the function (3.113) is increasing for all σ > 0.

It means that the increase of the control never could be compensated by the decay of the

solution, the control of L-R method no longer converges. In fact, with the same method in

Theorem 3.2.1, it is easy to show that, system (3.109) is not null controllable when α = 1/2.

To prove Theorem 3.8.1, we need the following Lemma, which is the analogue of Lemma

3.3.1:

Lemma 3.8.1. Let α > 1/2. Let T and C > 0 be two positive constants and t be

sufficiently small. Then function

f(σ) = eC√

σ(1 + σαt)−[ Tt

](3.113)


i). f(σ) is decreasing in the interval(( C

αT )2/(2α−1), ( αTCt)

2).

ii). f(σ) < e−δ√

σ in the interval((2C+δ

T )2/(2α−1), ( αT(C+δ)t )

2)

for any δ > 0.

iii). It holds

limt→0

f(σ)∣∣∣σ=( αT

Ct)2

= 0. (3.114)


g(x) = exp(Cx− [

T

t ] ln(1 + t x2α)).

The derivative of g(x) with respect to x reads

g′(x) = g(x)(C − [

T

t ]2αtx2α−1

1 + tx2α

).

Here g(x) is a decreasing function if and only if g′(x) is negative. To find the decreasing

part of f(σ) we need to compute the interval in which g′(x) has negative value. Since

g(x) > 0, it is sufficient that

Ctx2α − 2αTx2α−1 + C < 0. (3.115)


Figure 3.1: Figure of ϕ(y)

Denote tx by y. It is obvious from (3.115) that y = tx should satisfy

ϕ(y)= y2α − 2αTy2α−1 + C(t)2α−1 < 0.

Figure 1 is the graph of ϕ(y), with A0 = (0, C(t)2α−1), B0 = (2αTC , C(t)2α−1). We

assume that A = (δ0, 0) and B = (δ1, 0) be the two points of intersection of ϕ(y) and the

y-axis.

For t sufficiently small, we compute A and B, respectively.

1). Point A. Obviously, δ0 tends to 0 as t tends to 0. Hence, by

ϕ(δ0) = Cδ2α0 − 2αTδ2α−1

0 + C(t)2α−1 = 0,

we have

δ0 =( C

2αT

)1/(2α−1)t+ o(t). (3.116)

2). Point B. Obviously δ1 tends to 2αTC as t tends to 0. Let δ1 = 2αT

C + δ and we

compute

ϕ(δ1) = C(2αT

C+ δ)2α − 2αT (

2αT

C+ δ)2α−1 + C(t)2α−1 = 0,

⇔ 0 =(2αT

C

)2α−1δ + o(δ) + (t)2α−1,

⇔ δ = −(Ct

2αT

)2α−1+ o((t)2α−1

).

3.8. TIME-DISCRETE FRACTIONAL ORDER PARABOLIC EQUATIONS 65

Hence, we arrive at

δ1 =2αT

C−(Ct

2αT

)2α−1+ o((t)2α−1

). (3.117)

Let (x0, x1) be the interval in which g′(x) has negative value. Taking into account that

y = tx, we have

x0 =( C

2αT

)1/(2α−1)+O(t) < (

C

αT)1/(2α−1),

x1 =2αT

Ct −(Ct

2αT

)2α−2+ o((t)2α−2

)>

αT

Ct .

Hence we conclude that This finishes the proof of i) by replacing x as√σ.

Now we consider the function H(x) = f(x2)eδ√

σ. H(x) < 1 if and only if

lnH(x) = C x− [T

t ] ln(1 + t x2α) + δ x < 0.


lnH(x) = (C + δ)x − [T

t ]t x2α +

[ Tt ](t)2

2x4α −

[ Tt ](t)3x6α

3(1 + ξt x2α)3,

for some ξ ∈ (0,t x2α). Since x > 0, we deduce that lnH(x) < 0 if

C + δ − (T −t)x2α−1 +Tt

2x4α−1 < 0. (3.118)

We claim that (3.118) is satisfied when x ∈(( 2C+δ

T−t)1/(2α−1), ( 2C

Tt)1/(4α−1)

). This is due to

the fact that

C + ε− (T −t)x2α−1 +Tt

2x4α−1 < −C +

Tt2

x4α−1 < 0.

Similarly, it is easy to show that H(x) is decreasing for x ∈((C+δ

T )1/(2α−1), αT(C+δ)t

).

Hence, H(x) < 1 when x satisfies

(2C + δ

T)1/(2α−1) < x < max

((

2C

Tt)1/(4α−1),

αT

(C + δ)t). (3.119)

Moreover, since t is sufficiently small, we get that the right side of (3.119) equals to

αT(C+δ)t and consequently H(x) < 1 in the interval

(2C + δ

T)1/(2α−1) < x <

αT

(C + δ)t .


Hence, we finish to prove ii) by replacing x as√σ.

Next, by

limt→0

f(σ)∣∣∣σ=( αT

Ct)2

= limt→0

exp(αT − [ T

t ]t ln(1 + α2T 2

C2t)

t)

= 0, (3.120)

we get (3.114).

Now we prove Theorem 3.8.1.

Sketch of the Proof of Theorem 3.8.1: The proof is similar to that in section 5.

The only difference is that the function (3.36) is replaced by (3.113) and the corresponding

Lemma is replaced by Lemma 3.8.1.

Chapter 4

The time-discrete wave equation

4.1. Introduction

In this chapter, we are interested in the time semi-discretization of systems (2.11) and

(2.13).

To begin with, we present the time-discrete schemes of continuous systems. As the

same in the previous chapter, we denote by yk and uk respectively the approximations of

the solution y and the control u of (2.11) at time tk = kt for any k = 0, · · · ,K. We then

introduce the following trapezoidal time semi-discretization of (2.11):

yk+1 + yk−1 − 2yk

(t)2 − ∆

(yk+1 + yk−1

2

)= 0,

in Ω, k = 1, · · · ,K − 1

yk+1 + yk−1

2= ukχΓ0 , on Γ, k = 1, · · · ,K − 1

y0 = y0, y1 = y0 + ty1, in Ω.

(4.1)

Here (y0, y1) ∈ L2(Ω)×H−1(Ω) are the data given in system (2.11) that allow determining

the initial data for the time-discrete system too. We refer to Theorem 4.4.2 below for the

well-posedness of system (4.1) by means of the transposition method.

The controllability problem for system (4.1) may be formulated as follows: For any

(y0, y1) ∈ L2(Ω)×H−1(Ω), to find a control uk ∈ L2(Γ0)k=1,··· ,K−1 such that the solution

ykk=0,··· ,K of (4.1) satisfies:

yK−1 = yK = 0 in Ω. (4.2)

Note that (4.2) is equivalent to the condition yK−1 = (yK −yK−1)/t = 0 that is a natural

discrete version of (2.12).

67

68 CHAPTER 4. THE TIME-DISCRETE WAVE EQUATION

As in the context of the above continuous wave equation, we also consider the uncon-

trolled system

ϕk+1 + ϕk−1 − 2ϕk

(t)2 − ∆

(ϕk+1 + ϕk−1

2

)= 0,

in Ω, k = 1, · · · ,K − 1

ϕk = 0, on Γ, k = 0, · · · ,KϕK = ϕt

0 + tϕt1 , ϕK−1 = ϕt

0 , in Ω,

(4.3)

where (ϕt0 , ϕt

1 ) ∈ (H10 (Ω))2. In particular, to guarantee the convergence of the solutions

of (4.3) towards those of (2.13) one considers convergent data such thatϕt

0 → ϕ0 strongly in H10 (Ω),

ϕt1 → ϕ1 strongly in L2(Ω).

as K → ∞ (or t→ 0), (4.4)

with tϕt1 being bounded in H1

0 (Ω). Obviously because of the density of H10 (Ω) in L2(Ω)

this choice is always possible.

We emphasize that the choice of the values of ϕK and ϕK−1 in (4.3) is motivated by

the transposition arguments that are needed to define the solution of the time-discrete

non-homogenous problem (4.1), as we will see in Section 4.8.

The energy of system (4.3) is given by

Ekt

=

1

2

∫

Ω

(∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

+|∇ϕk+1|2 + |∇ϕk|2

2

)dx, k = 0, · · · ,K − 1, (4.5)

which is a discrete counterpart of the continuous energy E(t) in (2.15). Multiplying the

first equation of system (4.3) by (ϕk+1 −ϕk−1)/2 and integrating it in Ω, using integration

by parts, it is easy to show the following property of conservation of energy:

Ekt = E0

t, k = 0, · · · ,K − 1. (4.6)

Consequently the scheme under consideration is stable and its convergence (in the classical

sense of numerical analysis) is guaranteed in an appropriate functional setting (in particular

in the finite-energy space H10 (Ω) × L2(Ω), under the condition (4.4)).

By means of classical duality arguments, it is easy to show that the above controllability

property (4.2) is equivalent to the following boundary observability property for solutions

ϕkk=0,··· ,K of (4.3):

E0t ≤ Ct

K−1∑

k=1

∫

Γ0

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ0, ∀ (ϕt0 , ϕt

1 ) ∈ (H10 (Ω))2. (4.7)

4.1. INTRODUCTION 69

The analysis of controllability and/or observability properties of numerical approxima-

tion schemes for the wave equation has been the object of intensive studies. However most

analytical results concern the case of space semi-discretizations (see [110] and the refer-

ences cited therein). In practical applications, fully discrete schemes need to be used. The

most typical example is the classical central scheme which converges under a suitable CFL

condition ([24, 25, 93]). However, in the present setting in which the Laplacian ∆ is kept

continuous, without discretizing it, this scheme is unsuitable since it is unstable. To see

this, we choose µ2jj≥1 to be eigenvalues of the Dirichlet Laplacian and Φj∞j=1 ⊂ H1

0 (Ω)

the corresponding eigenvectors (constituting an orthonormal basis of L2(Ω)), i.e.,

−∆Φj = µ2jΦj, in Ω

Φj = 0, on Γ.(4.8)

Since µ2jj≥1 tends to infinity, it is easy to check that the central scheme

ϕk+1 + ϕk−1 − 2ϕk

(t)2 − ∆ϕk = 0 (4.9)

is unstable. Indeed, the stability of (4.9) would be equivalent to the stability of the scheme

ϕk+1 + ϕk−1 − 2ϕk

(t)2 + µ2jϕ

k = 0

for all values of µ2j , j ≥ 1. But this stability property fails clearly, regardless how small

t is, when µ2j is large enough. Hence, we choose the trapezoidal scheme (4.3) for the

time-discrete problem, which is stable (due to the property of conservation of energy), as

mentioned before.

Similar to Chapter 3, the first result of this chapter is of negative nature. Indeed, as

we shall see in Theorem 4.5.1, the observability inequality (4.7) (resp. the controllability

property (4.2)) fails for system (4.3) (resp. (4.1)) without filtering. From the proof of

Theorem 4.5.1 below, it will be obvious that these negative results of observability and

controllability are related to the fact that the spaces in which the solutions evolve are infinite

dimensional; while the number of time-steps is finite. Accordingly, to make the observability

inequality possible one has to restrict the class of solutions of the adjoint system (4.3) under

consideration by filtering the high frequency components. Similarly, since the property of

exact controllability of system (4.1) fails, the final requirement (4.2) has to be relaxed

by considering only low frequency projections of the solutions. This filtering method has

been applied successfully in the previous chapter in the context of controllability of time-

discrete heat equations (see also [105]) and space semi-discrete schemes for wave equations

([5, 32, 108, 110]).


As far as we know, the subject of control and observation of the time-discrete wave

equation under consideration has not been addressed before. In this chapter we shall develop

a discrete version of the classical multiplier approach which allows to view the time-discrete

wave equation as an evolution process with its own dynamics.

As in the continuous case, the multiplier technique we use here applies mainly to the

case when the controller/observer Γ0 is given in (2.17) and some variants ([83]), but does not

work when (T,Ω,Γ0) is assumed to satisfy the GCC. As we shall see, the main advantage

of our multiplier approach is that the filtering parameter we use has the optimal scaling in

what concerns the frequency of observed/controlled solutions with respect to t.It is important to note that this kind of results can not be obtained by standard per-

turbation arguments that rely simply on measuring the distance between solutions of the

time-discrete and continuous wave equations. Indeed, when proceeding that way, one needs

much stronger filtering requirements. In other words, the optimal filtering can only be ob-

tained by a careful analysis of the time evolution of the system under consideration. This

is already well-known in the context of space semi-discretizations (see [110]). Our discrete

multiplier approach can also be extended to other PDEs of conservative nature, and in

particular to the Schrodinger, plate, Maxwell’s equations, among others.

The rest of the chapter is organized as follows. In Section 4.2, we collect some prelimi-

nary results which are useful in what follows. In Section 4.3, we present two fundamental

identities by means of discrete multipliers, which will play an important role in the sequel.

In Section 4.4 we discuss the hidden regularity property of solutions of (4.3) and the uni-

form well-posedness property of system (4.1). Section 4.5 is devoted to show the lack of

controllability/observability of systems (4.1) and (4.3) without filtering. The uniform ob-

servability result for (4.3) is presented in Section 4.6. In Section 4.7 we show the optimality

of the filtering parameter in the uniform observability result. Moreover, we give a heuristic

explanation of the necessity of the filtering in terms of the group velocity of propagation of

waves. Finally, Section 4.8 is devoted to the uniform controllability of system (4.1) and the

convergence of the controls and solutions.

4.2. Preliminaries

In this section, we collect some preliminary results that will be used in the sequel.

First of all, for any given fk ∈ L2(Ω)k=1,··· ,K−1 and gk ∈ H10 (Ω)k=1,··· ,K with

4.2. PRELIMINARIES 71

g1 = gK = 0, suppose θk ∈ H10 (Ω)k=0,··· ,K solves the system

θk+1 + θk−1 − 2θk

(t)2 − ∆

(θk+1 + θk−1

2

)= fk +

gk+1 − gk

t ,

in Ω, k = 1, · · · ,K − 1

θk = 0, on Γ, k = 0, · · · ,K.

(4.10)

We define the energy of system (4.10) by

Ekt

=

1

2

∫

Ω

(∣∣∣∣θk+1 − θk

t

∣∣∣∣2

+|∇θk+1|2 + |∇θk|2

2

)dx. (4.11)

We establish the following discrete version of the energy estimate:

Lemma 4.2.1. For any t > 0, it holds

max0≤k≤K−1

Ekt ≤ C

min(E0t, EK−1

t

)+

[t

K−1∑

k=1

(|fk|L2(Ω) + |gk|H1

0 (Ω)

)]2

+tK−1∑

k=1

∫

Ω

∣∣∣∣fk g

k+1 + gk

2

∣∣∣∣ dx.

(4.12)

Proof of Lemma 4.2.1: Fix any ℓ ∈ 1, · · ·K − 1. Multiplying both sides of (4.10)

by (θk+1 − θk−1)/2t, integrating it in Ω and summing it for k = 1, · · · , ℓ, we obtain:

tℓ∑

k=1

∫

Ω

[θk+1 + θk−1 − 2θk

(t)2 − ∆

(θk+1 + θk−1

2

)]θk+1 − θk−1

2t dx

= tℓ∑

k=1

∫

Ω

(fk +

gk+1 − gk

t

)θk+1 − θk−1

2t dx.

(4.13)

From the definition of the energy Ekt in (4.11), the left hand side term of (4.13) can be

written as

LHS of (4.13) =

ℓ∑

k=1

(Ekt − Ek−1

t

)= Eℓ

t − E0t. (4.14)

We now analyze the right hand side (4.13). It is clear that

∣∣∣∣∣tℓ∑

k=1

∫

Ωfk θ

k+1 − θk−1

2t dx

∣∣∣∣∣ ≤ Ctℓ∑

k=1

|fk|L2(Ω)

(√Ekt +

√Ek−1t

). (4.15)


On the other hand, since g1 = 0, it follows

tℓ∑

k=1

∫

Ω

gk+1 − gk

tθk+1 − θk−1

2t dx =

ℓ∑

k=1

∫

Ω(gk+1 − gk)

(θk+1 − θk

2t +θk − θk−1

2t

)dx

=

ℓ∑

k=1

∫

Ωgk+1 θ

k+1 − θk

2t dx−ℓ∑

k=1

∫

Ωgk θ

k − θk−1

2t dx

+ℓ∑

k=1

∫

Ωgk+1 θ

k − θk−1

2t dx−ℓ∑

k=1

∫

Ωgk θ

k+1 − θk

2t dx

=

∫

Ωgℓ+1 θ

ℓ+1 − θℓ

2t dx+ℓ∑

k=2

∫

Ωgk θ

k − θk−1

2t dx−ℓ−1∑

k=0

∫

Ωgk+1 θ

k+1 − θk

2t dx

+

∫

Ωgℓ+1 θ

ℓ − θℓ−1

2t dx+ℓ−1∑

k=1

∫

Ωgk+1 θ

k − θk−1

2t dx−ℓ∑

k=1

∫

Ωgk θ

k+1 − θk

2t dx

=

∫

Ωgℓ+1 θ

ℓ+1 − θℓ

t dx−tℓ∑

k=1

∫

Ω

gk+1 + gk

2

θk+1 + θk−1 − 2θk

(t)2 dx.

(4.16)

It is obvious that

∣∣∣∣∫

Ωgℓ+1 θ

ℓ+1 − θℓ

t dx

∣∣∣∣ ≤ C|gℓ+1|L2(Ω)

√Eℓt ≤ C|gℓ+1|H1

0 (Ω)

√Eℓt. (4.17)

In view of (4.10) and noting again g1 = 0, it holds

−tℓ∑

k=1

∫

Ω

gk+1 + gk

2

θk+1 + θk−1 − 2θk

(t)2 dx

= −tℓ∑

k=1

∫

Ω

gk+1 + gk

2

[∆

(θk+1 + θk−1

2

)+ fk +

gk+1 − gk

t

]dx

= tℓ∑

k=1

∫

Ω

[∇(gk+1 + gk

2

)· ∇(θk+1 + θk−1

2

)

+fk gk+1 + gk

2

]dx−

∫

Ω

∣∣gℓ+1∣∣2 −

∣∣g1∣∣2

2dx

≤Chℓ∑

k=1

(|gk+1|H1

0 (Ω) + |gk|H10 (Ω)

)(|θk+1|H1

0 (Ω) + |θk−1|H10 (Ω)

)+ t

ℓ∑

k=1

∫

Ωfk g

k+1 + gk

2dx

≤ Ch

ℓ∑

k=1

(|gk+1|H1

0 (Ω) + |gk|H10 (Ω)

)(√Ekt +

√Ek−1t

)+ t

ℓ∑

k=1

∫

Ωfk g

k+1 + gk

2dx.

(4.18)


Combining (4.13)–(4.18) , we conclude that

Eℓt ≤ Ch

ℓ∑

k=1

(|fk|L2(Ω) + |gk+1|H1

0 (Ω) + |gk|H10 (Ω)

)(√Ekt +

√Ek−1t

)

+tℓ∑

k=1

∫

Ω

∣∣∣∣fk g

k+1 + gk

2

∣∣∣∣ dx+ E0t.

(4.19)

Put

F ℓt

= max

0≤k≤ℓEkt. (4.20)

Since (4.19) holds for all ℓ = 1, · · · ,K − 1, it is still true if Eℓt is replaced by F ℓ

t. Hence,

from (4.19) and recalling g1 = gK = 0, we obtain

F ℓt ≤ Ch

K−1∑

k=1

(|fk|L2(Ω) + |gk|H1

0 (Ω)

)√F ℓt + t

ℓ∑

k=1

∫

Ω

∣∣∣∣fk g

k+1 + gk

2

∣∣∣∣ dx+ E0t

≤ C

[t

K−1∑

k=1

(|fk|L2(Ω) + |gk|H1

0 (Ω)

)]2

+F ℓt

2+ t

K−1∑

k=1

∫

Ω

∣∣∣∣fk g

k+1 + gk

2

∣∣∣∣ dx+ E0t.

(4.21)

Now, combining (4.20) and (4.21), it follows

max1≤k≤K−1

Ekt ≤ C

E0t +

[t

K−1∑

k=1

(|fk|L2(Ω) + |gk|H1

0 (Ω)

)]2

+tK−1∑

k=1

∫

Ω

∣∣∣∣fk g

k+1 + gk

2

∣∣∣∣ dx. (4.22)

Noting the “time reversibility” of system (4.10), similar to (4.22), we have

max1≤k≤K−1

Ekt ≤ C

EK−1t +

[t

K−1∑

k=1

(|fk|L2(Ω) + |gk|H1

0 (Ω)

)]2

+tK−1∑

k=1

∫

Ω

∣∣∣∣fk g

k+1 + gk

2

∣∣∣∣ dx. (4.23)

Finally, combining (4.22) and (4.23), we end up with the desired estimate (4.12).

Next, we claim that the solution of system (4.3) can be expressed explicitly by means

of Fourier series. Indeed, we have


Lemma 4.2.2. Assume ϕK =

∞∑

j=1

(aj + tbj)Φj and ϕK−1 =

∞∑

j=1

ajΦj (or, equivalently,

ϕt0 =

∞∑

j=1

ajΦj and ϕt1 =

∞∑

j=1

bjΦj). Then the solution of system (4.3) is given by

ϕk =

∞∑

j=1

eiωj(K−k−1) (e

iωj − 1)aj −tbj2i sinωj

+ e−iωj(K−k−1) (1 − e−iωj)aj + tbj2i sinωj

Φj,

(4.24)

where

ωj = arccos1

1 + (t)2µ2j/2

. (4.25)

Remark 4.2.1. i) From (4.24), it is easy to see that, if for some j0 ∈ lN, the data ϕK and

ϕK−1 belong to spanΦj | j ≤ j0, then the same is true for ϕk for all 1 ≤ k ≤ K.

ii) From (4.24), one deduces also that, if aj and bj are chosen so that (e−iωj −1)aj = hbj

(resp. (eiωj − 1)aj = hbj) for j = 1, 2, · · · , then

ϕk =∞∑

j=1

ajeiωj(K−k−1)Φj (resp. ϕk =

∞∑

j=1

aje−iωj(K−k−1)Φj).

Proof of Lemma 4.2.2: By Lemma 4.2.1, it suffices to find a solution ϕk of the form

ϕk =∞∑

j=1

rkj Φj (4.26)

such that, for j = 1, 2, · · · ,

rk+1j + rk−1

j − 2rkj

(t)2 + µ2j

rk+1j + rk−1

j

2= 0, k = 1, 2, · · · ,K − 1, (4.27)

and

rKj = aj + hbj , rK−1

j = aj . (4.28)

The characteristic polynomial of (4.27) (which is a difference equation) reads

p(λ)=λ2 + 1 − 2λ

(t)2 + µ2j

λ2 + 1

2.

The roots nj and mj of p(λ) are as follows

nj =1 + ihµj

√1 + (t)2µ2

j/4

1 + (t)2µ2j/2

, mj =1 − ihµj

√1 + (t)2µ2

j/4

1 + (t)2µ2j/2

. (4.29)


Therefore, the rkj ’s satisfy

rk+1j − njr

kj = mj(r

kj − njr

k−1j ).

By induction, the unique rkj satisfying (4.27)–(4.28) is given by (recalling (4.28))

rkj =

(nj)K−k − (mj)

K−k

nj −mjrK−1j − (nj)

K−k−1 − (mj)K−k−1

nj −mjrKj

=1

nj −mj

(nj)

K−k−1 [(nj − 1)aj −tbj] − (mj)K−k−1 [(mj − 1)aj −tbj]

.

(4.30)

Noting the definition of ωj in (4.25), by (4.29), it follows

nj = eiωj , mj = e−iωj .

Therefore, the rkj ’s given by (4.30) can be re-written as

rkj = eiωj(K−k−1) (e

iωj − 1)aj −tbj2i sinωj

+ e−iωj(K−k−1) (1 − e−iωj )aj + tbj2i sinωj

. (4.31)

Finally, combining (4.26) and (4.31), we conclude the desired formula (4.24).

The third one is a classical multiplier identity for the Dirichlet Laplacian:

Lemma 4.2.3. Let = (1, · · · , d) ∈ C1(Ω; lRd). Then, for any ψ ∈ H2(Ω) ∩H10 (Ω), it

holds

∫

Ω · ∇ψ∆ψdx =

1

2

[∫

Γ · ν

∣∣∣∣∂ψ

∂ν

∣∣∣∣2

dΓ +

∫

Ωdiv|∇ψ|2dx

]−

d∑

i,j=1

∫

Ωi

xjψxiψxjdx. (4.32)

Identity (4.32) can be easily proved multiplying ∆ψ by · ∇ψ where · stands for the

scalar product in lRd. We refer to [57, identity (1.25)] or to [102, Lemma 3.3] for the details.

Finally, following [57, pp. 8–9], one has

Lemma 4.2.4. For any f ∈ L2(Ω) and g ∈ H10 (Ω), it holds

∣∣∣∣∫

Ωf

[(x− x0) · ∇g +

d− 1

2g

]dx

∣∣∣∣ ≤R

2

∫

Ω

(f2 + |∇g|2

)dx, (4.33)

where R is as in (2.17).


4.3. Identities via multipliers

This section is addressed to establish two fundamental identities by means of discrete

multipliers. First, we show the following one:

Lemma 4.3.1. Let = (1, · · · , d) ∈ C1(Ω; lRd). Then, for any t > 0, any fk ∈L2(Ω)k=1,··· ,K−1, any gk ∈ H1

0 (Ω)k=1,··· ,K with g1 = gK = 0, and any θk ∈ H2(Ω) ∩H1

0 (Ω)k=0,··· ,K satisfying (4.10), it holds

t2

K−1∑

k=1

∫

Γ · ν

∣∣∣∣∂

∂ν

(θk+1 + θk−1

2

)∣∣∣∣2

dΓ = U + V1 − V2 −W, (4.34)

where

U =

∫

Ω ·[∇(θK + θK−2

2

)θK − θK−1

t −∇(θ2 + θ0

2

)θ1 − θ0

t

]dx

+

∫

Ω ·[∇(θK−1 − θK−2

2

)θK − θK−1

t + ∇(θ2 − θ1

2

)θ1 − θ0

t

]dx,

(4.35)

V1 = tK−1∑

k=1

∫

Ωdiv

(θk+1 − θk)(θk − θk−1)

2(t)2 dx, (4.36)

V2 =t2

K−1∑

k=1

∫

Ω

[div

∣∣∣∣∇(θk+1 + θk−1

2

)∣∣∣∣2

−2d∑

i,j=1

ixj

(θk+1 + θk−1

2

)

xi

(θk+1 + θk−1

2

)

xj

dx,(4.37)

W = tK−1∑

k=1

∫

Ω · ∇

(θk+1 + θk−1

2

)fkdx

+t2

K−1∑

k=2

∫

Ω

(θk+1 − θk

t +θk−1 − θk−2

t

)( · ∇gk + divgk

)dx.

(4.38)

Proof of Lemma 4.3.1: Multiplying (4.10) by · ∇(θk+1 + θk−1)/2 (which is a

discrete version of the multiplier ·∇θ for the wave equation), integrating it in Ω, summing

it for k = 1, · · · ,K − 1, it follows

tK−1∑

k=1

∫

Ω · ∇

(θk+1 + θk−1

2

)θk+1 + θk−1 − 2θk

(t)2 dx

−tK−1∑

k=1

∫

Ω · ∇

(θk+1 + θk−1

2

)∆

(θk+1 + θk−1

2

)dx

= tK−1∑

k=1

∫

Ω · ∇

(θk+1 + θk−1

2

)(fk +

gk+1 − gk

t

)dx.

(4.39)

4.3. IDENTITIES VIA MULTIPLIERS 77

First, we analyze the first term in the left hand side of (4.39):

tK−1∑

k=1

∫

Ω · ∇

(θk+1 + θk−1

2

)θk+1 + θk−1 − 2θk

(t)2 dx

=K−1∑

k=1

∫

Ω · ∇

(θk+1 + θk−1

2

)(θk+1 − θk

t − θk − θk−1

t

)dx

=

∫

Ω ·[

K∑

k=2

∇(θk + θk−2

2

)θk − θk−1

t −K−1∑

k=1

∇(θk+1 + θk−1

2

)θk − θk−1

t

]dx

= −K−1∑

k=2

∫

Ω · ∇

(θk+1 − θk + θk−1 − θk−2

2

)θk − θk−1

t dx

+

∫

Ω ·[∇(θK + θK−2

2

)θK − θK−1

t −∇(θ2 + θ0

2

)θ1 − θ0

t

]dx.

(4.40)

However,

−K−1∑

k=2

∫

Ω · ∇

(θk+1 − θk + θk−1 − θk−2

2

)θk − θk−1

t dx

= −∫

Ω ·[

K−1∑

k=2

∇(θk+1 − θk

2

)θk − θk−1

t +

K−2∑

k=1

∇(θk − θk−1

2

)θk+1 − θk

t

]dx

=

∫

Ω ·[∇(θK−1 − θK−2

2

)θK − θK−1

t + ∇(θ2 − θ1

2

)θ1 − θ0

t

]dx

−K−1∑

k=1

∫

Ω · ∇

[(θk+1 − θk)(θk − θk−1)

2t

]dx.

(4.41)

Noting that

−K−1∑

k=1

∫

Ω · ∇

[(θk+1 − θk)(θk − θk−1)

2t

]dx =

K−1∑

k=1

∫

Ωdiv

(θk+1 − θk)(θk − θk−1)

2t dx,

from (4.40)–(4.41), it follows

tK−1∑

k=1

∫

Ω · ∇

(θk+1 + θk−1

2

)θk+1 + θk−1 − 2θk

(t)2 dx = U + V1, (4.42)

where U and V1 are defined respectively by (4.35) and (4.36).

Next, we analyze the second term in the left hand side of (4.39). Applying Lemma 4.2.3

(with ψ replaced by (θk+1 + θk−1)/2), we find

tK−1∑

k=1

∫

Ω · ∇

(θk+1 + θk−1

2

)∆

(θk+1 + θk−1

2

)dx

=t2

K−1∑

k=1

∫

Γ · ν

∣∣∣∣∂

∂ν

(θk+1 + θk−1

2

)∣∣∣∣2

dΓ + V2,

(4.43)


where V2 is defined by (4.37).

Further, using integration by parts and noting g1 = gK = 0, it follows

tK−1∑

k=1

∫

Ω · ∇

(θk+1 + θk−1

2

)gk+1 − gk

t dx

= −tK−1∑

k=1

∫

Ω

θk+1 + θk−1

2

[ · ∇

(gk+1 − gk

t

)+ div

gk+1 − gk

t

]dx

= −K∑

k=2

∫

Ω

θk + θk−2

2

( · ∇gk + div gk

)dx

+K−1∑

k=1

∫

Ω

θk+1 + θk−1

2

( · ∇gk + div gk

)dx

=t2

K−1∑

k=2

∫

Ω

(θk+1 − θk

t +θk−1 − θk−2

t

)( · ∇gk + divgk

)dx.

(4.44)

Finally, by (4.39), (4.42)–(4.44) and recalling the definition of W in (4.38), we conclude

the desired identity (4.34).

As we shall see in the next section, Lemma 4.3.1 is the basis to provide an important

hidden regularity property of solutions of system (4.10), and via which the well-posedness

of system (4.1) follows. Meanwhile, as a consequence of Lemma 4.3.1, we now show the

following identity for the solutions of (4.3), which will play a crucial role in the proof of

Theorem 4.6.1:

Lemma 4.3.2. For any t > 0 and any solution ϕkk=0,··· ,K of (4.3), it holds

t2

K−1∑

k=0

∫

Ω

(∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

+|∇ϕk+1|2 + |∇ϕk|2

2

)dx+X + Y + Z

=t2

K−1∑

k=1

∫

Γ(x− x0) · ν

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ,

(4.45)

where

X =

∫

Ω

[(x− x0) · ∇

(ϕK + ϕK−2

2

)+d− 1

2ϕK

]ϕK − ϕK−1

t dx

−∫

Ω

[(x− x0) · ∇

(ϕ2 + ϕ0

2

)+d− 1

2ϕ0

]ϕ1 − ϕ0

t dx,

(4.46)


Y =d

2

[(t)2

K−1∑

k=1

∫

Ω∆

(ϕk+1 + ϕk−1

2

)ϕk − ϕk−1

t dx−t∫

Ω

∣∣∣∣ϕK − ϕK−1

t

∣∣∣∣2

dx

]

+

∫

Ω(x− x0)·

[∇(ϕK−1 − ϕK−2

2

)ϕK − ϕK−1

t + ∇(ϕ2 − ϕ1

2

)ϕ1 − ϕ0

t

]dx,

(4.47)

Z =(d− 2)t

8

K−1∑

k=1

∫

Ω

∣∣∣∇(ϕk+1 − ϕk−1)∣∣∣2dx− (d− 1)t

4

K−1∑

k=0

∫

Ω

∣∣∣∇(ϕk+1 − ϕk

)∣∣∣2dx

−(d− 1)t4

∫

Ω

(∇ϕK · ∇ϕK−1 + ∇ϕ1 · ∇ϕ0

)dx

+(d− 2)t

4

∫

Ω

(|∇ϕK−1|2 + |∇ϕ1|2

)dx.

(4.48)

Remark 4.3.1. Identity (4.45) is a time-discrete analogue of the well-known identity for

the wave equation (4.3) obtained by multipliers, which reads (see [57]):

1

2

∫ T

0

∫

Ω

[|ϕt|2 + |∇ϕ|2

]dxdt+ X =

1

2

∫ T

0

∫

Γ(x− x0) · ν

∣∣∣∂ϕ

∂ν

∣∣∣2dΓdt. (4.49)

Here,

X =

∫

Ω

[(x− x0) · ∇ϕ+

d− 1

2ϕ

]ϕtdx

∣∣∣T

t=0. (4.50)

There are clear analogies between (4.45) and (4.49). In fact the only major differences are

that, in the discrete version (4.45), two extra reminder terms (Y and Z) appear, which

are due to the time discretization. It is easy to see, formally, that Y and Z tend to zero as

t→ 0. But this convergence does not hold uniformly for all solutions. Consequently, these

added terms impose the need of using filtering of the high frequencies to obtain observability

inequalities and also modify the observability time, as we shall see.

Proof of Lemma 4.3.2: We use Lemma 4.3.1 with = x−x0, fk = 0 (k = 1, · · · ,K−

1), gk = 0 (k = 1, · · · ,K) and θk = ϕk (k = 0, · · · ,K). Clearly, in this case W = 0 (recall

(4.38) for W ).

For V1 defined in (4.36) (with θk replaced by ϕk), noting div = d and using the first


equation in (4.3), one has

V1 = d

K−1∑

k=1

∫

Ω

(ϕk+1 − ϕk)(ϕk − ϕk−1)

2t dx

= dK−1∑

k=1

∫

Ω

[(ϕk+1 + ϕk−1 − 2ϕk) + (ϕk − ϕk−1)

]ϕk − ϕk−1

2t dx

=dh

2

K−1∑

k=1

∫

Ω

∣∣∣∣ϕk − ϕk−1

t

∣∣∣∣2

dx+dh2

2

K−1∑

k=1

∫

Ω

(ϕk+1 + ϕk−1 − 2ϕk

(t)2)ϕk − ϕk−1

t dx

=dh

2

[K−1∑

k=0

∫

Ω

∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

dx−∫

Ω

∣∣∣∣ϕK − ϕK−1

t

∣∣∣∣2

dx

]

+dh2

2

K−1∑

k=1

∫

Ω∆

(ϕk+1 + ϕk−1

2

)ϕk − ϕk−1

t dx.

(4.51)

For V2 defined in (4.37), noting ixj

= δij (the Kronecker delta) and using the elementary

identity (a+ b)2 = 2(a2 + b2) − (a− b)2 for any a, b ∈ lR, it follows

V2 =(d− 2)t

2

K−1∑

k=1

∫

Ω

∣∣∣∣∇(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dx

=(d− 2)t

2

K−1∑

k=1

∫

Ω

[|∇ϕk+1|2 + |∇ϕk−1|2

2−∣∣∣∣∇(ϕk+1 − ϕk−1

2

)∣∣∣∣2]dx

=(d− 2)t

2

K−1∑

k=1

∫

Ω

|∇ϕk+1|22

dx+K−2∑

k=0

∫

Ω

|∇ϕk|22

dx

−K−1∑

k=1

∫

Ω

∣∣∣∣∇(ϕk+1 − ϕk−1

2

)∣∣∣∣2

dx

=(d− 2)t

2

K−1∑

k=0

∫

Ω

|∇ϕk+1|2 + |∇ϕk|22

dx−∫

Ω

|∇ϕK−1|2 + |∇ϕ1|22

dx

−K−1∑

k=1

∫

Ω

∣∣∣∣∇(ϕk+1 − ϕk−1

2

)∣∣∣∣2

dx

.

(4.52)

Now, by (4.34) in Lemma 4.3.1, recalling the definition of U in (4.35) (with θk replaced


by ϕk), noting W = 0 and (4.52), we conclude that

t2

K−1∑

k=0

∫

Ω

(∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

+|∇ϕk+1|2 + |∇ϕk|2

2

)dx

+

∫

Ω(x− x0) ·

[∇(ϕK + ϕK−2

2

)ϕK − ϕK−1

t −∇(ϕ2 + ϕ0

2

)ϕ1 − ϕ0

t

]dx

+(d− 1)t

2

K−1∑

k=0

∫

Ω

(∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

− |∇ϕk+1|2 + |∇ϕk|22

)dx+ Y

=t2

K−1∑

k=1

∫

Γ(x− x0) · ν

∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

) ∣∣∣2dΓ

−(d− 2)t2

[∫

Ω

|∇ϕK−1|2 + |∇ϕ1|22

dx+K−1∑

k=1

∫

Ω

∣∣∣∇(ϕk+1 − ϕk−1

2

) ∣∣∣2dx

],

(4.53)

where Y is defined in (4.47).

On the other hand, multiplying the first equation of (4.3) by ϕk (which is a discrete

version of the multiplier ϕ in the time-continuous setting, that leads to the identity of

equipartition of energy), integrating it in Ω, summing it for k = 1, · · · ,K − 1 and using

integration by parts, we obtain:

0 = tK−1∑

k=1

∫

Ω

[ϕk+1 + ϕk−1 − 2ϕk

(t)2 − ∆

(ϕk+1 + ϕk−1

2

)]ϕkdx

=

K−1∑

k=1

∫

Ω

[(ϕk+1 − ϕk)ϕk

t − (ϕk − ϕk−1)ϕk

t

]dx

−tK−1∑

k=1

∫

Ω∆

(ϕk+1 + ϕk−1

2

)ϕkdx

=

K−1∑

k=1

∫

Ω

(ϕk+1 − ϕk)ϕk

t dx−K−2∑

k=0

∫

Ω

(ϕk+1 − ϕk)ϕk+1

t dx

−tK−1∑

k=1

∫

Ω∆

(ϕk+1 + ϕk−1

2

)ϕkdx

= −tK−1∑

k=0

∫

Ω

∣∣∣ϕk+1 − ϕk

t∣∣∣2dx+

∫

Ω

[(ϕK − ϕK−1)ϕK

t − (ϕ1 − ϕ0)ϕ0

t

]dx

+tK−1∑

k=1

∫

Ω∇(ϕk+1 + ϕk−1

2

)· ∇ϕkdx.

(4.54)


However

tK−1∑

k=1

∫

Ω∇(ϕk+1 + ϕk−1

2

)· ∇ϕkdx

= tK−1∑

k=1

∫

Ω|∇ϕk|2dx+ t

K−1∑

k=1

∫

Ω

[∇(ϕk+1 − ϕk

2

)−∇

(ϕk − ϕk−1

2

)]· ∇ϕkdx

= tK−1∑

k=0

∫

Ω

|∇ϕk+1|2 + |∇ϕk|22

dx− t2

K−1∑

k=0

∫

Ω

∣∣∣∇(ϕk+1 − ϕk)∣∣∣2dx

−t2

∫

Ω

(∇ϕK · ∇ϕK−1 + ∇ϕ1 · ∇ϕ0

)dx.

(4.55)

Combining (4.54) and (4.55), we end up with the following equipartition of energy identity

for the time semi-discrete system (4.3):

tK−1∑

k=0

∫

Ω

(∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

− |∇ϕk+1|2 + |∇ϕk|22

)dx

=

∫

Ω

[(ϕK − ϕK−1)ϕK

t − (ϕ1 − ϕ0)ϕ0

t

]dx

−t2

K−1∑

k=0

∫

Ω

∣∣∣∇(ϕk+1 − ϕk)∣∣∣2dx− t

2

∫

Ω

(∇ϕK · ∇ϕK−1 + ∇ϕ1 · ∇ϕ0

)dx.

(4.56)

Finally, substituting (4.56) into (4.53) and recalling (4.46) and (4.48) respectively for

the definition of X and Z, we conclude the desired identity (4.45).

4.4. Hidden regularity and well-posedness

This section is devoted to show a hidden regularity property of solutions of system (4.10)

and to establish the well-posedness of system (4.1).

We begin with the following hidden regularity property of solutions of system (4.10)

(recall (4.11) for the definition of Ekt):

Theorem 4.4.1. For any t > 0, any fk ∈ L2(Ω)k=1,··· ,K−1, any gk ∈ H10 (Ω)k=1,··· ,K

with g1 = gK = 0 in Ω, and any θk ∈ H10 (Ω)k=0,··· ,K satisfying (4.10), it holds

tK−1∑

k=1

∫

Γ

∣∣∣∣∂

∂ν

(θk+1 + θk−1

2

)∣∣∣∣2

dΓ

≤ C

min(E0t, EK−1

t

)+

[t

K−1∑

k=1

(|fk|L2(Ω) + |gk|H1

0 (Ω)

)]2

+ tK−1∑

k=1

∫

Ω

∣∣∣∣fk g

k+1 + gk

2

∣∣∣∣ dx

.

(4.57)

4.4. HIDDEN REGULARITY AND WELL-POSEDNESS 83

Remark 4.4.1. When t tends to zero, the limit of the system (4.10) is

θtt − ∆θ = f + gt, in (0, T ) × Ω

θ = 0, in (0, T ) × Γ.(4.58)

Inequality (4.57) is a time-discrete analogue of the following boundary estimate of (4.58):

∫ T

0

∫

Γ

∣∣∣∣∂ϕ

∂ν

∣∣∣∣2

dΓdt ≤ C

min

(∫

Ω

[|∇θ(0)|2 + |θt(0)|2

]dx,

∫

Ω

[|∇θ(T )|2 + |θt(T )|2

]dx

)

+[|f |L1(0,T ;L2(Ω)) + |g|L1(0,T ;H1

0 (Ω))

]2+

∫ T

0

∫

Ω|fg|dxdt

.

Proof of Theorem 4.4.1: As in [57], we choose a vector ∈ C1(Ω; lRd) so that = ν

on the boundary Γ. Then, the desired estimate (4.57) follows immediately from Lemma

4.3.1 and Lemma 4.2.1.

We now establish the well-posedness of system (4.1) by means of a discrete version of the

classical transposition approach ([57]). For this purpose, for any fk ∈ L2(Ω)k=1,··· ,K−1,

and any gk ∈ H10 (Ω)k=1,··· ,K with g1 = gK = 0, we consider the following adjoint problem

of system (4.1):

ζk+1 + ζk−1 − 2ζk

(t)2 − ∆

(ζk+1 + ζk−1

2

)= fk +

gk+1 − gk

t ,

in Ω, k = 1, · · · ,K − 1

ζk = 0, on Γ, k = 0, · · · ,KζK = ζK−1 = 0, in Ω.

(4.59)

It is easy to see that (4.59) admits a unique solution ζk ∈ H10 (Ω)k=0,··· ,K . By Theorem

4.4.1, this solution has the regularity property ∂∂ν

(ζk+1+ζk−1

2

)∈ L2(Γ), for k = 1, · · · ,K−1.

In order to give a reasonable definition for the solution of the non-homogenous boundary

problem (4.1) in terms of the transposition method, we consider first the case when the

control ukk=0,··· ,K and the initial data (y0, y1) are sufficiently smooth. The following

result holds:

Lemma 4.4.1. Assume that yk ∈ H2(Ω)k=0,··· ,Ksatisfies (4.1). Then

tK−1∑

k=1

∫

Ωfk y

k+1 + yk−1

2dx− t

2

K−1∑

k=2

∫

Ωgk

(yk+1 − yk

t +yk−1 − yk−2

t

)dx

=

∫

Ωζ0y

1 − y0

t dx−∫

Ω

ζ1 − ζ0

t y0dx−tK−1∑

k=1

∫

Γ0

∂

∂ν

(ζk+1 + ζk−1

2

)ukdΓ0.

(4.60)


Proof of Lemma 4.4.1: Multiplying both sides of (4.10) by (yk+1 +yk−1)/2, integrat-

ing the resulting identity in Ω, summing it for k = 1, · · · ,K − 1, one obtains:

tK−1∑

k=1

[∫

Ω

ζk+1 + ζk−1 − 2ζk

(t)2yk+1 + yk−1

2dx− ζk+1 + ζk−1

2∆

(yk+1 + yk−1

2

)

−∫

Γ0

∂

∂ν

(ζk+1 + ζk−1

2

)ukdΓ0

]

= tK−1∑

k=1

∫

Ω

(fk +

gk+1 − gk

t

)yk+1 + yk−1

2dx.

(4.61)

Recalling ζK = ζK−1 = 0 in Ω, it is clear that

tK−1∑

k=1

∫

Ω

ζk+1 + ζk−1 − 2ζk

(t)2yk+1 + yk−1

2dx

= tK−1∑

k=1

∫

Ω

ζk+1 + ζk−1

2

yk+1 + yk−1

(t)2 dx−tK−1∑

k=1

∫

Ω

ζkyk+1 + ζkyk−1

(t)2 dx

=

∫

Ωζ0 y

1 − y0

t dx−∫

Ω

ζ1 − ζ0

t y0dx+ tK−1∑

k=1

∫

Ω

ζk+1 + ζk−1

2

yk+1 + yk−1 − 2yk

(t)2 dx.

(4.62)

Also, noting g1 = gK = 0 in Ω, it holds

tK−1∑

k=1

∫

Ω

gk+1 − gk

tyk+1 + yk−1

2dx = −t

2

K−1∑

k=2

∫

Ωgk

(yk+1 − yk

t +yk−1 − yk−2

t

)dx.

(4.63)

Finally, from (4.61)–(4.63) and noting that yk ∈ H2(Ω)k=0,··· ,K satisfy the first equa-

tion in (4.1), the desired identity (4.60) follows.

Note that (4.60) still makes sense even if the regularity of ykk=0,··· ,K is relaxed as

follows

yk+1 + yk−1 ∈ L2(Ω), k = 1, · · · ,K − 1,

yk+1 − yk

t +yk−1 − yk−2

t ∈ H−1(Ω), k = 2, · · · ,K − 1.(4.64)

This is consistent with the existence result for (2.11) (in terms of the transposition method).

Indeed, under the condition u ∈ L2(Γ × (0, T )) it is well-known that the solution of (2.11)

satisfies y ∈ C([0, T ];L2(Ω))∩C1([0, T ];H−1(Ω)). Note that formally, letting t→ 0, (4.64)

leads to y(t, ·) ∈ L2(Ω) and yt(t, ·) ∈ H−1(Ω). This observation motivates the definition of

solution for system (4.1).

More precisely, set

H =ykk=0,··· ,K

∣∣∣ y0, · · · , yK satisfy (4.64). (4.65)

4.4. HIDDEN REGULARITY AND WELL-POSEDNESS 85

We introduce the following

Definition 4.4.1. We say ykk=0,··· ,K ∈ H to be a solution of (4.1), in the sense of

transposition, if y0 = y0, y1 = y0 + hy1, and for any fk ∈ L2(Ω)k=1,··· ,K−1, and gk ∈

H10 (Ω)k=1,··· ,K with g1 = gK = 0, it holds

tK−1∑

k=1

∫

Ωfk y

k+1 + yk−1

2dx− t

2

K−1∑

k=2

⟨gk,

yk+1 − yk

t +yk−1 − yk−2

t

⟩

H10 (Ω),H−1(Ω)

=⟨ζ0, y1

⟩H1

0 (Ω),H−1(Ω)−∫

Ω

ζ1 − ζ0

t y0dx−tK−1∑

k=1

∫

Γ0

∂

∂ν

(ζk+1 + ζk−1

2

)ukdΓ0,

(4.66)

where ζk ∈ H10 (Ω)k=0,··· ,K is the unique solution of (4.59).

We now show the following well-posedness result for this system:

Theorem 4.4.2. Assume (y0, y1) ∈ L2(Ω) ×H−1(Ω) and uk ∈ L2(Γ0)k=1,··· ,K−1. Then

system (4.1) admits one and only one solution ykk=0,··· ,K ∈ H in the sense of Definition

4.4.1. Moreover,

i) When K is odd,(y2ℓ, y2ℓ+1−y2ℓ

t

)∈ L2(Ω) ×H−1(Ω) for ℓ = 0, 1, · · · , [K2 ], and

maxℓ=0,1,··· ,[ K

2]

∥∥∥∥(y2ℓ,

y2ℓ+1 − y2ℓ

t

)∥∥∥∥2

L2(Ω)×H−1(Ω)

≤ C

(‖(y0, y1)‖2

L2(Ω)×H−1(Ω) + tK−1∑

k=1

∥∥∥uk∥∥∥

2

L2(Γ0)

).

(4.67)

ii) When K is even,(y2ℓ, y2ℓ−1−y2ℓ−2

t

)∈ L2(Ω) ×H−1(Ω) for ℓ = 1, · · · , K

2 , and

maxℓ=1,··· , K

2

∥∥∥∥(y2ℓ,

y2ℓ−1 − y2ℓ−2

t

)∥∥∥∥2

L2(Ω)×H−1(Ω)

≤ C

(‖(y0, y1)‖2

L2(Ω)×H−1(Ω) + tK−1∑

k=1

∥∥∥uk∥∥∥

2

L2(Γ0)

).

(4.68)

Furthermore, the constant C > 0 in the estimates (4.67) and (4.68) is independent of the

time-step t.

Proof of Theorem 4.4.2: The proof is standard, and hence we give only a sketch.

First of all, by Lemma 4.2.1, Theorem 4.4.1 and using the usual duality argument (e.g.


[57]), we conclude that system (4.1) admits a solution ykk=0,··· ,K ∈ H in the sense of

Definition 4.4.1, which verifies

maxk=1,··· ,K−1

∥∥∥yk+1 + yk−1∥∥∥

2

L2(Ω)+ max

k=2,··· ,K−1

∥∥∥∥yk+1 − yk

t +yk−1 − yk−2

t

∥∥∥∥2

H−1(Ω)

≤ C

(‖(y0, y1)‖2

L2(Ω)×H−1(Ω) + tK−1∑

k=1

∥∥∥uk∥∥∥

2

L2(Γ0)

).

(4.69)

Inequality (4.4.1) implies the uniqueness of the solution of system (4.1). On the other hand,

the constant C in this estimate is independent of t.Next, we show the “regularity” properties (4.67)–(4.68) for solution ykk=0,··· ,K . For

this purpose, for any ℓ ∈ 0, 1, · · · , [K2 ], we choose the test functions fk and gk in (4.66)

as follows:

fk =

(−1)(k+3)/2f1, for k = 1, 3, · · · , 2ℓ− 1

0, for k = 2, 4, · · · , 2ℓ, 2ℓ + 1, 2ℓ+ 2, · · · ,K − 1,

where f1 is arbitrary, and gk ≡ 0 for all k = 1, · · · ,K. Now, by Lemma 4.2.1, Theorem 4.4.1

and using the usual duality argument again, similar to (4.69), one deduces that y2ℓ − y0 ∈L2(Ω), and

∥∥∥y2ℓ − y0∥∥∥

2

L2(Ω)≤ Ch

K−1∑

k=1

∥∥∥uk∥∥∥

2

L2(Γ0),

via which (and noting that y0 ∈ L2(Ω)) the boundedness of each of y2,y4,...y2ℓ in L2(Ω)

follows (with a bound which is independent of the time step t). Similarly, noting that

y1 ∈ H−1(Ω), one obtains the rest result in (4.67).

4.5. Lack of controllability/observability without filtering

This section is devoted to prove the following negative controllability/observability re-

sult:

Theorem 4.5.1. For any given t > 0 and any nonempty open subset Γ0 of Γ, system

(4.3) is not observable, and therefore, system (4.1) is not null controllable.

Proof of Theorem 4.5.1: We emphasize that, in this proof, t is fixed so that

the system under consideration involves only a finite number of time-steps while it is a

distributed parameter system (infinite-dimensional one) in space. This is precisely the

main reason for the lack of observability results. The proof is divided into two steps.

4.5. LACK OF CONTROLLABILITY/OBSERVABILITY WITHOUT FILTERING 87

Step 1. We first show that inequality (4.7) fails for system (4.3) when Γ0 = Γ. For

this, put

fn =

n∑

j=1

|µj |−d/2Φj.

(Recall that d is the dimensions of Ω). By Weyl’s formula ([26]), µk ∼ C(Ω)k1/d as k → ∞.

Therefore,

||fn||2L2(Ω) =

n∑

j=1

|µj|−d → ∞, as n→ ∞; (4.70)

while, fnn≥1 is bounded in H−s(Ω) for all s > 0.

It is obvious that fn ∈ H2(Ω) ∩ H10 (Ω) for any n. We choose the final data of (4.3)

to be (ϕKn , ϕ

K−1n ) = (fn, 0) and denote the corresponding solution by ϕk

nKK=0. Note that

ϕK−2n , · · · , ϕ0

n are inductively determined by the following iterative elliptic systems

ϕk−1n − 1

2(t)2∆ϕk−1

n = 2ϕkn − ϕk+1

n +1

2(t)2∆ϕk+1

n , k = K − 1, · · · 1. (4.71)

By standard elliptic regularity theory, it is easy to see that ϕkn ∈ H2(Ω) ∩ H1

0 (Ω) for any

n ∈ lN.

On the other hand, (4.71) can be rewritten as

ϕk+1n + ϕk−1

n − 1

2(t)2∆

(ϕk+1

n + ϕk−1n

)= 2ϕk

n, k = K − 1, · · · 1.

This, combined with the standard regularity theory for elliptic equations of second order,

gives

K−1∑

k=1

∥∥∥ϕk+1n + ϕk−1

n

∥∥∥H1

0 (Ω)≤ C(t)

K−1∑

k=1

∥∥∥ϕkn

∥∥∥H−1(Ω)

≤ C(t) ‖fn‖H−1(Ω) . (4.72)

One can also re-write (4.71) as

ϕk+1n + ϕk−1

n =2

(t)2 (−∆)−1(2ϕk

n −(ϕk+1

n + ϕk−1n

)), k = K − 1, · · · 1.

Therefore, using again the standard elliptic regularity theory, we conclude that for any

τ ≤ 2, it holds

K−1∑

k=1

||ϕk+1n + ϕk−1

n ||Hτ (Ω) ≤ C(t)||2ϕkn −

(ϕk+1

n + ϕk−1n

)||Hτ−2(Ω)

≤ C(t)(||fn||Hτ−2(Ω) + ||fn||H−1(Ω)).

(4.73)


Hence, for any given t > 0 and 3/2 < τ < 2, using trace theorem, it follows from (4.73)

that

tK−1∑

k=1

∫

Γ

∣∣∣∣∂

∂ν

(ϕk+1

n + ϕk−1n

2

)∣∣∣∣2

dΓ ≤ C(t)K−1∑

k=1

||ϕk+1n + ϕk−1

n ||2Hτ (Ω)

≤ C(t)||fn||2Hτ−2(Ω).

(4.74)

The energy E0t of (4.3) with data (ϕK

n , ϕK−1n ) = (fn, 0) reads

E0t = EK−1

t =1

2

∫

Ω

(∣∣∣fn

t∣∣∣2+

|∇fn|22

)dx ≥ 1

2(t)2 ‖fn‖2L2(Ω) . (4.75)

Now, recalling that fnn≥1 is bounded in H−s(Ω) for all s > 0, taking (4.70), (4.74) and

(4.75) into account, we obtain that

limn→∞

E0t

tK−1∑

k=1

∫

Γ

∣∣∣∣∂

∂ν

(ϕk+1

n + ϕk−1n

2

)∣∣∣∣2

dΓ

= ∞. (4.76)

Thus, (4.7) fails. Consequently, system (4.3) is not observable (even when Γ0 = Γ).

Step 2. We now show that system (4.1) is not null controllable by means of a con-

tradiction argument. Assume that for any (y0, y1) ∈ L2(Ω) × H−1(Ω), there is a control

uk ∈ L2(Γ0)k=1,··· ,K−1 such that the solution ykk=0,··· ,K of (4.1) satisfies the null-

controllability property (4.2). The control is not unique, and therefore, we choose the one

of minimal norm. By the closed graph theorem, we deduce that

K−1∑

k=1

|uk|L2(Γ0) ≤ C|(y0, y1)|L2(Ω)×H−1(Ω). (4.77)

Multiplying the first equation of system (4.1) by (ϕk+1 + ϕk−1)/2, integrating it in Ω,

summing it for k = 1, · · · ,K − 1 and noting Theorem 4.4.2, it follows

⟨ϕ0, y1

⟩H1

0 (Ω),H−1(Ω)−∫

Ω

ϕ1 − ϕ0

t y0dx = tK−1∑

k=1

∫

Γ0

∂

∂ν

(ϕk+1 + ϕk−1

2

)ukdΓ0. (4.78)

Combining (4.77) and (4.78), one deduces easily that inequality (4.7) holds. From Step 1,

this is a contradiction.

4.6. Uniform observability under filtering

In this section, we shall establish uniform observability estimates for system (4.3) (with

respect to the time step t) after filtering the spurious high frequency components.

4.6. UNIFORM OBSERVABILITY UNDER FILTERING 89

4.6.1. Statement of the uniform observability result

As mentioned in Introduction, due to the negative results stated in Theorem 4.5.1, we

need to introduce suitable filtering spaces in which the solutions of system (4.3) evolve.

Recalling the definition of C1,s, C0,s, C−1,s in (1.4), (1.5), (1.6) and the corresponding pro-

jections π1,s, π0,s, π−1,s on H10 (Ω), L2(Ω) and H−1(Ω), our uniform observability result for

system (4.3) is stated as follows:

Theorem 4.6.1. Let T > 2R. Then there exist two constants t0 > 0 and δ > 0, depending

only on d, T and R, such that for all (ϕt0 , ϕt

1 ) ∈ C1,δt−2 × C0,δt−2, the corresponding

solution ϕkk=0,··· ,K of (4.3) satisfies

E0t ≤ Ct

K−1∑

k=1

∫

Γ0

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ0, (4.79)

for all t ∈ (0,t0].

Remark 4.6.1. In the proof we see that δ depends only on d, T and R. In particular

it indicates that δ decreases as T decreases. This is natural since, as T decreases, less

and less time-step iterations are involved in system (4.3) and, consequently, less Fourier

components of the solutions may be observed. Further, δ tends to zero as T tends to 2R.

This is natural too since our proof of (4.79) is based on the method of multipliers which

works at the continuous level for all T > 2R but that, at the time-discrete level, due to the

added dispersive effects, may hardly work when T is very close to 2R, except if the filtering

is strong enough.

Remark 4.6.2. In view of the hidden regularity result of Theorem 4.4.1, the right hand

side term of (4.79) is finite.

Remark 4.6.3. In the observability result of Theorem 4.6.1, the filtering parameter has

been taken to be of the order of t−2. This is the optimal order for the filtering parameter

since for higher frequencies there are solutions for which the observability constant blows-

up, as Theorem 4.7.1 in the next section shows. However, as we shall see, the necessity

of the filtering parameter δ to be small is not completely justified. In fact, our analysis

of the velocity of propagation of solutions in section 4.7 supports that, whatever δ > 0 is,

observability could be expected to hold for large enough values of time T .


4.6.2. A technical result

As mentioned before, the key point in the proof of Theorem 4.6.1 is Lemma 4.3.2. We

need to estimate first the term X and the error terms Y and Z in (4.45).

The following lemma provides an estimate on the term X + Y + Z:

Lemma 4.6.1. Let K be an integer, s > 0 and T > 0. Then for any (ϕt0 , ϕt

1 ) ∈ C1,s×C0,s,

the corresponding solution ϕkk=0,··· ,K of (4.3) satisfies

X + Z + Y ≥ −[2R+ a1t+ 3R

√st+ T

(d2

√sh+ a2s(t)2

) ]E0

t, (4.80)

where

a1 = 3d− 2 + max

(d− 1

2, 2

), a2 = min

(1, (2 − d)+

)+d− 1

2. (4.81)

Proof: The proof is divided in three steps, in which we estimateX,Y and Z, separately.

Note that, in view of the Fourier decomposition of solutions (see (4.24) in Remark 4.2.1),

the filtering introduced in the initial data is kept for all discrete time-steps k so that

∫

Ω|∇ϕk|2dx ≤ s

∫

Ω|ϕk|2dx

for all k = 0, · · · ,K and all solutions under consideration. This inequality will be used

throughout the proof.

Step 1. First, let us consider X. We have

∣∣∣∣d− 1

2

∫

Ω

ϕK − ϕK−2

2

ϕK − ϕK−1

t dx

∣∣∣∣

≤ (d− 1)t2

(∫

Ω

∣∣∣∣ϕK − ϕK−2

2t

∣∣∣∣2

dx

∫

Ω

∣∣∣∣ϕK − ϕK−1

t

∣∣∣∣2

dx

)1/2

≤ (d− 1)t2

(∫

Ω

∣∣ϕK − ϕK−1∣∣2 +

∣∣ϕK−1 − ϕK−2∣∣2

2(t)2 dx

)1/2(∫

Ω

∣∣∣∣ϕK − ϕK−1

t

∣∣∣∣2

dx

)1/2

≤ (d− 1)tE0t.

(4.82)

Further, applying Lemma 4.2.4 (with f and g replaced by (ϕK − ϕK−1)/t and (ϕK +


ϕK−2)/2), recalling (4.5) for the definition of Ekt and using (4.6), it follows

∣∣∣∣∫

Ω

[(x− x0) · ∇

(ϕK + ϕK−2

2

)+d− 1

2

(ϕK + ϕK−2

2

)]ϕK − ϕK−1

t dx

∣∣∣∣

≤ R

2

∫

Ω

[∣∣∣∣ϕK − ϕK−1

t

∣∣∣∣2

+

∣∣∣∣∇(ϕK + ϕK−2

2

)∣∣∣∣2]dx

≤ R

2

∫

Ω

(∣∣∣∣ϕK − ϕK−1

t

∣∣∣∣2

+|∇ϕK |2 + |∇ϕK−1|2

2+

|∇ϕK−2|2 − |∇ϕK−1|22

)dx

= RE0t +M,

(4.83)

where

M =Rh

4

∫

Ω∇(ϕK−2 + ϕK−1

)· ∇(ϕK−2 − ϕK−1

t

)dx.

Further, we estimate

M ≤ Rh

4

∫

Ω

∣∣∇(ϕK−2 + ϕK−1

)∣∣2 dx∫

Ω

∣∣∣∣∇(ϕK−2 − ϕK−1

t

)∣∣∣∣2

dx

1/2

≤ Rh

2

√s

∫

Ω

∣∣∇ϕK−2∣∣2 +

∣∣∇ϕK−1∣∣2

2dx

1/2∫

Ω

∣∣∣∣ϕK−2 − ϕK−1

t

∣∣∣∣2

dx

1/2

≤ t2

√sRE0

t.

(4.84)

Combining (4.82), (4.83) and (4.84) we obtain

∣∣∣∣∫

Ω

[(x− x0) · ∇

(ϕK + ϕK−2

2

)+d− 1

2ϕK

]ϕK − ϕK−1

t dx

∣∣∣∣

=

∣∣∣∣∫

Ω

[(x− x0) · ∇

(ϕK + ϕK−2

2

)+d− 1

2

(ϕK + ϕK−2

2

)]ϕK − ϕK−1

t dx

+d− 1

2

∫

Ω

ϕK − ϕK−2

2

ϕK − ϕK−1

t dx

∣∣∣∣

≤[R+ (d− 1)t+

hR

2

√s

]E0

t.

(4.85)

Similarly,

∣∣∣∣−∫

Ω

[(x− x0) · ∇

(ϕ2 + ϕ0

2

)+d− 1

2ϕ0

]ϕ1 − ϕ0

t dx

∣∣∣∣

≤[R+ (d− 1)t+

hR

2

√s

]E0

t.

(4.86)

Therefore, by (4.85)–(4.86) and recalling the definition of X in (4.46), we conclude that

|X| ≤[2R + 2(d − 1)t+Rt√s

]E0

t. (4.87)


Step 2. Next, let us consider Y . Using (4.6) and noting (4.24) in Remark 4.2.1, we

obtain

∣∣∣∣∫

Ω(x− x0) · ∇

(ϕK−1 − ϕK−2

2

)ϕK − ϕK−1

t dx

∣∣∣∣

≤ Rt2

[∫

Ω

∣∣∣∣∇(ϕK−1 − ϕK−2

t

)∣∣∣∣2

dx

∫

Ω

∣∣∣∣ϕK − ϕK−1

t

∣∣∣∣2

dx

]1/2

≤ Rt √s2

(∫

Ω

∣∣∣∣ϕK−1 − ϕK−2

t

∣∣∣∣2

dx

)1/2(∫

Ω

∣∣∣∣ϕK − ϕK−1

t

∣∣∣∣2

dx

)1/2

≤ Rt √sE0t.

(4.88)

Similarly, ∣∣∣∣∫

Ω(x− x0) · ∇

(ϕ2 − ϕ1

2

)ϕ1 − ϕ0

t dx

∣∣∣∣ ≤ Rt √sE0t. (4.89)

Further,

∣∣∣∣∣d(t)2

2

K−1∑

k=1

∫

Ω∆

(ϕk+1 + ϕk−1

2

)ϕk − ϕk−1

t dx

∣∣∣∣∣

=

∣∣∣∣∣d((t)2

2

K−1∑

k=1

∫

Ω∇(ϕk+1 + ϕk−1

2

)· ∇(ϕk − ϕk−1

t

)dx

∣∣∣∣∣

≤ d((t)22

K−1∑

k=1

[∫

Ω

∣∣∣∣∇(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dx

∫

Ω

∣∣∣∣∇(ϕk − ϕk−1

t

)∣∣∣∣2

dx

]1/2

≤ dt √s T2

E0t.

(4.90)

Also, ∣∣∣∣∣−dh

2

∫

Ω

∣∣∣∣ϕK − ϕK−1

t

∣∣∣∣2

dx

∣∣∣∣∣ ≤ dhE0t. (4.91)

By (4.88)–(4.91) and recalling the definition of Y in (4.47), we conclude that

|Y | ≤ t[d

(√s T

2+ 1

)+ 2R

√s

]E0

t. (4.92)

Step 3. Finally, we consider Z. It follows

tK−1∑

k=1

∫

Ω

∣∣∣∇(ϕk+1 − ϕk−1)∣∣∣2dx ≤ st3

K−1∑

k=1

∫

Ω

∣∣∣∣ϕk+1 − ϕk−1

t

∣∣∣∣2

dx

≤ 2st3K−1∑

k=1

∫

Ω

(∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

+

∣∣∣∣ϕk − ϕk−1

t

∣∣∣∣2)dx ≤ 8s(t)2TE0

t.

(4.93)


Since the first term in Z is nonnegative whenever d ≥ 2, we get from (4.93) that

(d− 2)t8

K−1∑

k=1

∫

Ω

∣∣∣∇(ϕk+1 − ϕk−1)∣∣∣2dx ≥

−sh2TE0

t, d = 1

0, d ≥ 2.(4.94)

Similarly,

tK−1∑

k=0

∫

Ω

∣∣∣∇(ϕk+1 − ϕk)∣∣∣2dx ≤ st3

K−1∑

k=0

∫

Ω

∣∣∣∣ϕk+1 − ϕk

t

∣∣∣∣2

dx ≤ 2s(t)2TE0t. (4.95)

Further,

−(d− 1)t4

∫

Ω∇ϕK · ∇ϕK−1dx+

(d− 2)t4

∫

Ω|∇ϕK−1|2dx

≥ −(d− 1)t16

∫

Ω|∇ϕK |2dx+

[(d− 2)t4

− (d− 1)t4

] ∫

Ω|∇ϕK−1|2dx

≥ −tmax

(d− 1

4, 1

)E0

t.

(4.96)

Similarly,

− (d− 1)t4

∫

Ω∇ϕ1 ·∇ϕ0dx+

(d− 2)t4

∫

Ω|∇ϕ1|2dx ≥ −tmax

(d− 1

4, 1

)E0

t. (4.97)

By (4.94)–(4.97), recalling the definition of Z in (4.48), we conclude that

Z ≥ −t[

min(1, (2 − d)+) +d− 1

2

]shT + max

(d− 1

2, 2

)E0

t. (4.98)

Now, combining (4.87), (4.92) and (4.98), we arrive at the desired estimate (4.80).

4.6.3. Proof of the uniform observability result

We are now in a position to prove the uniform observability result, i.e., Theorem 4.6.1.

Proof of Theorem 4.6.1: Combining (4.45) in Lemma 4.3.2 and (4.80) in Lemma

4.6.1, recalling the definition of Γ0 in (2.17), we deduce that

T(1 − d

2

√st− a2s(t)2

)−[2R+ a1t+ 3R

√sh]

E0t

≤ R

2t

K−1∑

k=1

∫

Γ0

∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

) ∣∣∣2dΓ0.

For this inequality to yield an estimate on E0t we need to choose s = δt−2 with t small

enough such that

a2δ +d

2

√δ < 1,


or, more precisely,

0 <√δ <

4√d2 + 16a2 + d

. (4.99)

Once this is done, for t ∈ (0,t0), T has to be chosen such that

T >2R + a1t0 + 3R

√δ

1 − d2

√δ − a2δ

≥ 2R. (4.100)

Hence, (4.79) holds for t ∈ (0,t0].Conversely, for any T > 2R one can always choose t0 and δ small enough so that (4.99)

and (4.100) hold and guaranteeing the uniform observability inequality. More precisely,

combining (4.99) and (4.100), the filtering parameter δ has to be chosen such that

0 < δ < min( 4√

d2 + 16a2 + d,

2a1T + 4R− 2T

a3 +√a2

3 − 4Ta2(a1T + 2R− T )

), (4.101)

with a3 = d2T + 3R.

4.7. Optimality of the filtering parameter

This section is addressed to analyze the optimality of the filtering mechanism introduced

in Theorem 4.6.1.

4.7.1. Optimality of the order of the filtering parameter

We first show the following result, which indicates that the order t−2 of the filtering

parameter that we have chosen in Theorem 4.6.1 is optimal.

Theorem 4.7.1. Assume Γ∗ is any nonempty open subset of Γ. Then, for any given a > 2,

it follows that

limt→0

sup(ϕt

0 ,ϕt1 )∈C1,t−a×C0,t−a

E0t

tK−1∑

k=1

∫

Γ∗

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ∗

= ∞. (4.102)

Proof of Theorem 4.7.1: Recall that Φj∞j=1 ⊂ H10 (Ω) denotes the orthonormal

basis of L2(Ω) constituted by the eigenvectors of the Dirichlet Laplacian and µ2jj≥1 the

corresponding eigenvalues. Since µj → +∞ as j → ∞, one can choose a j0 = j0(t) so

that t−a/2/2 ≤ µj0 ≤ t−a/2. In view of the fact that a > 2, this leads to

µj0t→ ∞, as t→ 0. (4.103)

4.7. OPTIMALITY OF THE FILTERING PARAMETER 95

Further, choose

ϕt0 =

1

µj0

Φj0, ϕt1 =

e−iωj0 − 1

µj0tΦj0, (4.104)

where ωj0 is defined by (4.25). One deduces that (ϕt0 , ϕt

1 ) ∈ C1,t−a × C0,t−a . Noting

the special choice of initial data in (4.104), by Lemma 4.2.2 (see also Remark 4.2.1 ii)), the

corresponding solution ϕkk=0,··· ,K of (4.3) is given by

ϕk =1

µj0

eiωj0(K−k−1)Φj0, k = 0, · · · ,K. (4.105)

Using (4.25), it follows

cos(ωj0) =2

2 + (µj0t)2, sin(ωj0) =

µj0t√

4 + (µj0t)22 + (µj0t)2

. (4.106)

Recalling the exact form of E0t in (4.5), combining (4.105) and (4.106), we compute

E0t = EK−1

t =1

2

∫

Ω

( |∇ϕK |2 + |∇ϕK−1|22

+∣∣∣ϕK − ϕK−1

t∣∣∣2)dx

=1

2

(1 +

∣∣∣e−iωj0 − 1

µj0t∣∣∣2)

=1

2

(1 +

∣∣∣cos(ωj0) − 1 − i sin(ωj0)

µj0t∣∣∣2)

=4 + (µj0t)2

2[2 + (µj0t)2].

(4.107)

On the other hand, via (4.105) and (4.106), one has

∫

Γ∗

∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣2dΓ∗

≤∣∣∣eiωj0 + e−iωj0

2µj0

∣∣∣2∫

Γ∗

∣∣∣∂Φj0

∂ν

∣∣∣2dΓ∗ ≤

cos2(ωj0)

µ2j0

∫

Γ

∣∣∣∂Φj0

∂ν

∣∣∣2dΓ.

(4.108)

We claim that ∫

Γ

∣∣∣∂Φj0

∂ν

∣∣∣2dΓ ≤ Cµ2

j0. (4.109)

Indeed, since Γ ∈ C2, one can find a 0 = (10, · · · , d

0) ∈ C1(Ω; lRd) such that 0 = ν on Γ

([57]). Applying Lemma 4.2.3 with = 0 and ψ = Φj0, we get

∫

Ω0·∇Φj0∆Φj0dx =

1

2

[∫

Γ

∣∣∣∂Φj0

∂ν

∣∣∣2dΓ +

∫

Ωdiv0|∇Φj0|2dx

]−

d∑

i,j=1

∫

Ω∂xj

i0∂xiΦj0∂xiΦj0dx.

Recall that ∆Φj0 = −µ2j0

Φj0 in Ω. Hence, (4.109) follows from

∫

Ω0 · ∇Φj0∆Φj0dx = −µ2

j0

∫

ΩΦj00 · ∇Φj0dx =

1

2µ2

j0

∫

Ωdiv0|Φj0|2dx ≤ Cµ2

j0,


and

d∑

i,j=1

∫

Ω∂xj

i0∂xiΦj0∂xiΦj0dx− 1

2

∫

Ωdiv0|∇Φj0|2dx ≤ C

∫

Ω|∇Φj0|2dx ≤ Cµ2

j0.

Combining (4.108) and (4.109), we find

tK−1∑

k=1

∫

Γ∗

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ∗ ≤ C cos2(ωj0). (4.110)

Finally, combining (4.107) and (4.110), and noting (4.103) and (4.106), it follows

sup(ϕt

0 ,ϕt1 )∈C1,t−a×C0,t−a

E0t

tK−1∑

k=1

∫

Γ∗

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ∗

≥ 4 + (µj0t)22[2 + (µj0t)2]

[2 + (µj0t)2]24C

=[4 + (µj0t)2][2 + (µj0t)2]

8C→ ∞ as t→ 0,

which gives (4.102).

Some remarks are in order.

Remark 4.7.1. The argument above, based on the use of separated variables monochro-

matic solutions, shows that the order of filtering µ2 ≤ Ch−2 is sharp, in the sense that the

observability inequality fails to be uniform when we take into account eigenvalues µ2 such

that µ2 ≫ t−2. Note however that our observability results require to restrict the class

of eigenvalues under consideration to µ2 ≤ δt−2 with δ > 0 small. The discussion above

does not justify the optimality of this smallness condition on the filtering constant. Actu-

ally, as we shall show in the next section, one may expect that uniform observability and

controllability properties hold within classes of filtered solutions of the form µ2 ≤ Ct−2

with arbitrary C > 0 for a sufficiently large time.

Remark 4.7.2. In a first look to this problem it might seem to be surprising that the

negative result in Theorem 4.7.1 is related to monochromatic waves. Nevertheless, the lack

of uniform observability is related to the fact that the quantity in the right hand side of

(4.110) is of the order of cos2(ωj0) which tends to zero as h → 0. Of course this does not

happen for the continuous wave equation. Indeed, if one computes for the solution for the

continuous-time wave equation (2.13) with initial data

ϕ0 =1

µj0

Φj0, ϕ1 =e−iωj0 − 1

µj0hΦj0,

4.7. OPTIMALITY OF THE FILTERING PARAMETER 97

the same as that in (4.104), one gets

ϕ =

[cos(µj0(T − t)) +

e−iωj0 − 1

µj0hsin(µj0(T − t))

]Φj0.

It is easy to check that the dominant corresponding term in the continuous-time boundary

observation∫ T0

∫Γ0

|∂ϕ/∂ν|2 dΓ0dt reads

∫ T

0cos2(µj0(T − t))dt.

Clearly, this term is bounded below (and therefore does not tend to zero) when h → 0,

contrarily to what happens for the corresponding discrete term cos2(ωj0).

4.7.2. A heuristic explanation

We now give a heuristic explanation of the necessity of filtering in terms of the group

velocity of propagation of the solutions of the time-discrete system (see [94, 110]). For doing

that we consider the time-discrete wave equation (4.3) in the whole space. Applying the

Fourier transform (the continuous one in space and the discrete one in time), we deduce

that the symbol of the time semi-discrete system (4.3) is

pt(τ, ξ) = −4 sin2 τt2

(t)2 + |ξ|2 cos(τt), (τ, ξ) ∈[− π

2t ,π

2t

]× lRd.

It is easy to see that, for all τ ∈[−π(2t)−1, π(2t)−1

], pt(τ, ξ) has two nontrivial roots

ξ± ∈ lRd. The bicharacteristic rays are defined as the solutions of the following Hamiltonian

system:

dx(s)

ds= 2ξ cos(τt), dt(s)

ds= −2 sin(τt)

t − |ξ|2t sin(τt),

dξ(s)

ds= 0,

dτ(s)

ds= 0.

As in the continuous case, the rays are straight lines. However, both the direction and the

velocity of propagation of the rays in this time-discrete setting case are different from the

time-continuous one.

Let us now illustrate the existence of bicharacteristic rays whose projection on lRd

propagates at a very low velocity or even does not move at all. For this, we fix any

x0 = (x0,1, · · · , x0,d) ∈ Ω and choose the initial time t0 = 0. Also, we choose the initial

microlocal direction (τ0, ξ0) = (τ0, ξ0,1, · · · , ξ0,d), as a root of Pt. Thus

|ξ0|2 =4 sin2 τ0t

2

(t)2 cos(τ0t), τ0 ∈

[− π

2t ,π

2t

].


Figure 4.1: The diagram of C(ξ). t = 0.1 (solid line) vs. t = 0.01 (dashed line). The

thick horizontal segment corresponds to the theoretical group velocity C(ξ) = 1 (in the

continuous case, i.e. t = 0).

Note that the above condition is satisfied for ξ0,1 = 2(t)−1 sin τ0t2 cos−1/2(τ0t) and

ξ0,2 = · · · = ξ0,d = 0, for instance. In this case we get

dx

dt=dx/ds

dt/ds= −cos3/2(τ0t)

cos τ0t2

and x′2(t) = · · · = x′d(t) = 0. Thus, xj(t) for j = 2, · · · , d remain constant and

x1(t) = x0,1 − t cos3/2(τ0t) cos−1 τ0t2

evolves with speed − cos3/2(τ0t) cos−1 τ0t2 , which tends to 0 when τ0t→ π

2−, or τ0t→

−π2

+. This allows us to show that, as t → 0, there exist rays that remain trapped on

a neighborhood of x0 for time intervals of arbitrarily large length. In order to guarantee

the boundary observability these rays have to be cut-off by filtering. This can be done by

restricting the Fourier spectrum of the solution to the range |τ | ≤ ρπ/2t with 0 < ρ < 1.

This corresponds to

|ξ|2 ≤ 4 sin2(ρπ/2)

(t)2 cos(ρπ/2), (4.111)

for the root of the symbol Pt.

This is the same scaling of the filtering operators we imposed on Theorems 4.6.1 and

4.8.1, namely, µ2j ≤ δ/(t)2. Note however that, in (4.111), as ρ→ 1, the filtering parameter

δ =4 sin2(ρπ/2)

cos(ρπ/2)→ ∞.

4.8. UNIFORM CONTROLLABILITY AND CONVERGENCE OF THE CONTROLS99

Thus, in principle, as mentioned above, the analysis of the velocity of propagation of bichar-

acteristic rays does not seem to justify the need of letting the filtering parameter δ small

enough as in Theorems 4.6.1 and 4.8.1. Thus, this last restriction seems to be imposed by

the rigidity of the method of multipliers rather than by the underlying wave propagation

phenomena.

We can reach similar conclusions by analyzing the behavior of the so-called group ve-

locity. Indeed, following [94], in 1 − d the group velocity has the form

C(ξ) =4

(2 + (t)2ξ2)√

4 + (t)2ξ2,

with the graphs as in Figure 1. Obviously, it tends to zero when (t)2ξ2 tends to infinity.

This corresponds precisely to the high frequency bicharacteristic rays constructed above

for which the velocity of propagation vanishes. Based on this analysis one can show that,

whatever the filtering parameter δ is, uniform observability requires the observation time

to be large enough with T (δ) ր ∞ as δ ր ∞. This may be done using an explicit

construction of solutions concentrated along rays (see, for instance, [61]). The positive

counterpart of this result guaranteeing that, for any value of the filtering parameter δ > 0,

uniform observability/controllability holds for large enough values of time, is an interesting

open problem whose complete solution will require the application of microlocal analysis

tools.

4.8. Uniform controllability and convergence of the controls

In this section, we present the following uniform partial controllability result for system

(4.1) and the convergence result for the controls :

Theorem 4.8.1. Let T , t0 and δ be given as in Theorem 4.6.1, and K > 1 be an odd

integer. Then for any t ∈ (0,t0] and any (y0, y1) ∈ L2(Ω) × H−1(Ω), there exists a

control uk ∈ L2(Γ0)k=1,··· ,K−1 such that the solution of (4.1) satisfies

i)

π0,δt−2yK−1 = π−1,δt−2yK − yK−1

t = 0 in Ω; (4.112)

ii) There exists a constant C > 0, independent of t, y0 and y1, such that

tK−1∑

k=1

∥∥∥uk∥∥∥

2

L2(Γ0)≤ C

∥∥∥∥(y0,

y1 − y0

t

)∥∥∥∥2

L2(Ω)×H−1(Ω)

; (4.113)


iii) When t→ 0,

Ut=

K−1∑

k=1

uk(x)χ[kt,(k+1)t)(t) −→ u strongly in L2((0, T ) × Γ0), (4.114)

where u is a control of system (2.11), fulfilling (2.12);

iv) When t→ 0,

yt= y0χ0(t) +

1

t

K−1∑

k=0

[(t− kt)yk+1 −

(t− (k + 1)t

)yk]χ(kt,(k+1)t](t)

−→ y strongly in C([0, T ];L2(Ω)) ∩H1([0, T ];H−1(Ω)),(4.115)

where y is the solution of system (2.11) with the control u as above.

The above theorem contains two results: the uniform partial controllability and the

convergence of the controls and states as t→ 0. The proof is standard. Indeed, the partial

controllability statement follows from Theorem 4.6.1 and classical duality arguments ([57]);

while for the convergence result, one may use the approach developed in [110]. However,

for readers’ convenience, we give below a sketch of the proof of Theorem 4.8.1.

Proof of Theorem 4.8.1: For any given T > 2R, choose a sufficiently small δ such

that Theorem 4.6.1 guarantees the uniform observability for (4.3). Recall that for any given

initial state (y0, y1) ∈ L2(Ω) ×H−1(Ω) of the continuous system (2.11), the initial data of

(4.1) are chosen to be (y0, y1−y0

t ) = (y0, y1).

For any (ϕt0 , ϕt

1 ) ∈ C1,δt−2 × C0,δt−2 , consider the functional

Jt(ϕt0 , ϕt

1 )= t

K−1∑

k=1

∫

Γ0

∣∣∣∣∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣2

dΓ0

−⟨y1, ϕ

0⟩H−1(Ω),H1

0 (Ω)+

∫

Ωy0ϕ1 − ϕ0

t dx,

where ϕkk=0,··· ,K is the solution of (4.3) with data (ϕt0 , ϕt

1 ). By Theorem 4.4.1,

Jt(ϕt0 , ϕt

1 ) is well-defined. Moreover Jt is convex, continuous and coercive in C1,δt−2×C0,δt−2, uniformly on t > 0. In view of Theorem 4.6.1, Jt(ϕ

t0 , ϕt

1 ) admits one and

only one minimizer (ϕt0 , ϕt

1 ) ∈ C1,δt−2 × C0,δt−2.

Let (ϕt0 , ϕt

1 ) be the minimizer of Jt(ϕt0 , ϕt

1 ) in C1,δt−2 × C0,δt−2 . It is easy to

4.8. UNIFORM CONTROLLABILITY AND CONVERGENCE OF THE CONTROLS101

check that

tK−1∑

k=1

∫

Γ0

∂

∂ν

(ϕk+1 + ϕk−1

2

)∂

∂ν

(ϕk+1 + ϕk−1

2

)dΓ0

=⟨y1, ϕ

0⟩H−1(Ω),H1

0 (Ω)−∫

Ωy0ϕ1 − ϕ0

t dx, ∀ (ϕt0 , ϕt

1 ) ∈ C1,δt−2 × C0,δt−2,

(4.116)

where ϕkk=0,··· ,K is the solution of system (4.3) with data (ϕt0 , ϕt

1 ).

Multiplying the first equation of system (4.1) by (ϕk+1 + ϕk−1)/2, integrating it in Ω,

summing it for k = 1, · · · ,K − 1 and noting Theorem 4.4.2, it follows

⟨ϕK−1,

yK − yK−1

t

⟩

H10 (Ω),H−1(Ω)

−∫

Ω

ϕk − ϕK−1

t yK−1dx

=⟨ϕ0, y1

⟩H1

0 (Ω),H−1(Ω)−∫

Ω

ϕ1 − ϕ0

t y0dx−tK−1∑

k=1

∫

Γ0

∂

∂ν

(ϕk+1 + ϕk−1

2

)ukdΓ0.

(4.117)

We now choose the control function ukk=1,··· ,K−1 in system (4.1) as follows

uk =∂

∂ν

(ϕk+1 + ϕk−1

2

)∣∣∣∣Γ0

, k = 1, · · · ,K − 1. (4.118)

Then, (4.116), (4.117),(4.118) together with form of the initial data in (2.13) yield

⟨ϕt

0 ,yK − yK−1

t

⟩

H10 (Ω),H−1(Ω)

−∫

Ωϕt

1 yK−1dx = 0, ∀ (ϕt0 , ϕt

1 ) ∈ C1,δt−2 ×C0,δt−2 .

This gives the controllability property (4.112). The desired estimate (4.113) follows imme-

diately from (4.118), (4.116) and Theorem 4.6.1.

Next, we prove the convergence of the controls. For this, recalling the exact form of

Ut in (4.114) and noting its uniform boundedness with K = 3, 5, · · · (which follows from

(4.113)), we conclude that, extracting subsequences, for some u ∈ L2((0, T ) × Γ0) and

(ϕ0, ϕ1) ∈ H10 (Ω) × L2(Ω),

Ut −→ u weakly in L2((0, T ) × Γ0),

(ϕt0 , ϕt

1 ) −→ (ϕ0, ϕ1) weakly in H10 (Ω) × L2(Ω).

as t→ 0. (4.119)

Moreover, one can show by standard arguments, that

u =∂ϕ

∂ν

∣∣∣∣(0,T )×Γ0

, (4.120)

where ϕ is the solution of (2.13) with data (ϕ0, ϕ1).


One can also use a classical Γ-convergence argument to show that the limit (ϕ0, ϕ1) is

the minimizer in H10 (Ω)× L2(Ω) of the functional J corresponding to the controllability of

the continuous wave equation.

Letting K → ∞ in (4.116), we deduce that ϕ satisfies

∫ T

0

∫

Γ0

∂ϕ

∂ν

∂ϕ

∂νdΓ0dt = 〈y1, ϕ(0)〉H−1(Ω),H1

0 (Ω) −∫

Ωy0ϕt(0)dx,

∀ (ϕ0, ϕ1) ∈ H10 (Ω) × L2(Ω),

(4.121)

where ϕ is the solution of (2.13) with data (ϕ0, ϕ1). Similar to the above, (4.121) implies

that the solution of system (2.11) with control u given by (4.120) satisfies (2.12).

On the other hand, by the weak convergence of (ϕt0 , ϕt

1 ) in H10 (Ω) × L2(Ω), recalling

the definition of Ut in (4.114), noting (4.118) and (4.120), we conclude from (4.116) and

(4.121) that ∫ T

0

∫

Γ0

|Ut|2dΓ0dt →∫ T

0

∫

Γ0

|u|2dΓ0dt as t→ 0. (4.122)

Combining (4.122) and the first convergence in (4.119), the desired strong convergence result

(4.114) follows.

Once the strong convergence of the controls is known, the estimates of Theorem 4.4.2

allow getting a uniform bound of ytt>0 (defined in (4.115)) in C([0, T ];L2(Ω)) ∩H1([0, T ];H−1(Ω)), which yields the desired strong convergence result for the extension

ytt>0 of the time-discrete solution ykk=0,··· ,K of (4.1) to continuous time, as indi-

cated by (4.115). This completes the proof of Theorem 4.8.1.

Chapter 5

Time-discrete conservative linear

systems

5.1. Introduction

In this chapter, we analyze the observability property of various time semi-discrete

schemes for system (2.18)-(2.19). As a consequence of our results, we present some applica-

tions to time-discrete schemes for wave, Schrodinger and KdV equations and fully discrete

approximations schemes for wave equations.

To begin with, we present a natural discretization of the continuous system. For any

t > 0, we denote by zk and yk respectively the approximations of the solution z and the

output function y of system (2.18)–(2.19) at time tk = kt for k ∈ Z. We then introduce

the following implicit midpoint time discretization of system (2.18):

zk+1 − zk

t = A(zk+1 + zk

2

), in X, k ∈ Z,

z0 given.

(5.1)

Consequently, the output function of (5.1) is given by

yk = Bzk, k ∈ Z. (5.2)

Taking into account that the spectrum of A is purely imaginary, it is easy to show that∥∥zk∥∥

X

is conserved in the discrete time variable k ∈ Z, i.e.∥∥zk∥∥

X=∥∥z0∥∥

X. Consequently the

scheme under consideration is stable and its convergence (in the classical sense of numerical

analysis) is guaranteed in an appropriate functional setting.

The uniform observability problem for system (5.1) is formulated as follows: To find a

103

104 CHAPTER 5. TIME-DISCRETE CONSERVATIVE LINEAR SYSTEMS

positive constant kT , independent of t, such that the solutions zk of system (5.1) satisfy:

kT

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥yk∥∥∥

2

Y, (5.3)

to all initial data z0 in an appropriate class.

Clearly, (5.3) is a discrete version of (2.21).

Note that this type of observability inequality appears naturally when dealing with

stabilization and controllability problems (see, for instance, [56], [95] and [110]). As we

explained in Introduction, for numerical approximation process, it is important that these

inequalities hold uniformly with respect to the discretization parameter(s) (here t only)

to recover uniform stabilization properties or the convergence of discrete controls to the

continuous ones. In the sequel, we are interested in describing which assumptions are

needed for inequality (5.3) to hold uniformly with respect to step size t. One expects to

do it so that, when letting t→ 0, one recovers the observability property of the continuous

model.

It can also be done by means of the spectral filtering mechanism. More precisely, re-

calling the definition of Cs in (1.38), we will prove that for many time discretizations of

(2.18)-(2.19) inequality (5.3) holds uniformly in the class Cδ/t for any δ > 0 and a suffi-

ciently large time Tδ.

One of the interesting applications of our results is that it allows us to develop a two-step

strategy to study the observability of fully discrete approximation schemes of (2.18)-(2.19).

First, taking into account the observability properties for space semi-discrete approximation

schemes, uniformly with respect to the space mesh-size parameter, as it has already been

done in many cases (see, for instance, [5], [32] and [75]). Second, from the results of this

Chapter on time discretizations, the observability inequality (with respect to the time and

space mesh-sizes) for the fully discrete approximation schemes is derived. See the details

in Section 5.5. To our knowledge, the observability issues for fully discrete approximation

schemes have been studied only in [73], in the very particular case of the 1-d wave equation.

The results we present here can be derived for a much wider class of systems and time-

discretizations schemes.

To complete our analysis of the discretizations of system (2.18)-(2.19), we also analyze

admissibility properties for the time semi-discrete systems throughout this Chapter. They

are useful when deriving controllability results out of the observability ones. More precisely,

5.2. THE IMPLICIT MID-POINT SCHEME 105

it allows proving controllability results by means of duality arguments combined with the

observability and admissibility results (see for instance the textbook [56] and the survey

article [110]). In particular, we prove that the admissibility inequality (2.20) can be inter-

preted in terms of the behavior of the wave packets. From this wave packet estimate, we

will deduce admissibility inequalities for the time semi-discrete schemes. This part can be

read independently from the last section.

The outline of this chapter is stated as follows.

In Section 5.2 we prove Theorem 5.2.2, from which we deduce the uniform observability

property (5.3) for system (5.1)-(5.2), assuming that the initial data are taken in some

subspace of filtered data Cδ/t for arbitrary δ > 0. Our proof of Theorem 5.2.2 is mainly

based on the resolvent estimate (5.9), combined with standard Fourier arguments adapted

to the time-discrete setting. In Section 5.3, we show how to apply Theorem 5.2.2 to obtain

similar results for time semi-discrete approximation schemes such as (5.34) and the Newmark

approximation schemes, for which we prove that a uniform observability inequality holds as

well, provided the initial data belong to Cδ/t. In Section 5.4, we give some applications to

the observability of some classical conservative equations, such as the Schrodinger equation

or the linearized KdV equation, etc. In Section 5.5, we give some applications of our main

results to fully discrete schemes for skew-adjoint systems as (2.18). In Section 5.6, we

present admissibility results similar to (2.20) for the time semi-discrete schemes used along

this chapter.

5.2. The implicit mid-point scheme

This section is devoted to the study of system (5.1)–(5.2). Let us first introduce some

notations and definitions.

The Hilbert space D(A) is endowed with the norm of the graph of A:

‖z‖21 = ‖z‖2

X + ‖Az‖2X .

It follows that B ∈ L(D(A), Y ) implies

‖Bz‖Y ≤ CB ‖Az‖X , ∀z ∈ D(A). (5.4)

We are now in position to claim the following theorem based on the resolvent estimate

(2.22):



Then, for any δ > 0, there exist Tδ and t0 > 0 such that for any T > Tδ and

t ∈ (0,t0), there exists a positive constant kT,δ, independent of t, such that the solution

zk of (5.1) satisfies

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥Bzk∥∥∥

2

Y, ∀ z0 ∈ Cδ/t. (5.5)

Moreover, Tδ can be taken to be such that

Tδ = π[M2(1 +

δ2

4

)2+m2C2

B

δ4

16

]1/2, (5.6)


Remark 5.2.1. If we filter at a scale smaller than t, for instance in the class Cδ/(t)α ,

with α < 1, then δ in (5.6) vanishes as t tends to zero. In that case the uniform observ-

ability time T0 we obtain is T0 = πM, which coincides with the estimate obtained by the

resolvent estimate (2.22) in the continuous setting (see [70]). Note that, however, even in

the continuous setting, in general πM is not the optimal observability time, although it is

the one one can get from the resolvent estimate method we employ in this chapter.

Proof of Theorem 5.2.1. It can be obtained as a consequence of the following theorem:

Theorem 5.2.2. Let δ > 0.

Assume that we have a sequence of vector spaces Xδ,t ⊂ X and a sequence of unbounded

operators (At, Bt) such that

(H1) For each t > 0, the operator At is skew-adjoint on Xδ,t, and the vector space

Xδ,t is globally invariant by At. Moreover,

‖Atz‖X ≤ δ

t ‖z‖X , ∀z ∈ Xδ,t, ∀t > 0. (5.7)

(H2) There exists a positive constant CB such that


(H3) There exist two positive constants M and m such that


Y ≥ ‖z‖2X ,

∀z ∈ Xδ,t ∪ D(At),∀ω ∈ lR, ∀t > 0.(5.9)


Then there exists a time Tδ such that for all time T > Tδ, there exists a constant kT,δ such

that for t small enough, the solution of

zk+1 − zk

t = At

(zk+1 + zk

2

), in Xδ,t, k ∈ Z, . (5.10)

with initial data z0 ∈ Xδ,t satisfies

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)


2

Y, ∀ z0 ∈ Xδ,t. (5.11)

Moreover, comparing to the optimal observability time T = πM in the continuous setting,

Tδ can be taken to be such that

Tδ = π[(

1 +δ2

4

)2M2 +m2C2

B

δ4

16

]1/2, (5.12)


Indeed, Theorem 5.2.2 provides an observability result within the class Cδ/t for system

(5.1)-(5.2), since one can easily verify that (H1)–(H3) hold by taking At = A, Bt = B

and Xδ,t = Cδ/t. ♦

Before getting into the proof of Theorem 5.2.2, let us first introduce the discrete Fourier

transform at scale t, which is one of the main ingredients of the proof of Theorem 5.2.2.

Definition 5.2.1. Given any sequence (uk) ∈ l2(tZ), we define its Fourier transform as:

u(τ) = t∑

k∈Z

uk exp(−iτkt), τt ∈ (−π, π]. (5.13)

For any function v ∈ L2(−π/t, π/t), we define the inverse Fourier transform at scale

t > 0:

vk =1

2π

∫ π/t

−π/tv(τ) exp(iτkt) dτ, k ∈ Z. (5.14)

According to Definition 5.2.1,

˜u = u, ˆv = v, (5.15)

and the Parseval’s identity holds

1

2π

∫ π/t

−π/t|u(τ)|2 dτ = t

∑

k∈Z

|uk|2. (5.16)

These properties will be used in the sequel.


Proof of Theorem 5.2.2. The proof is split into three parts.

Step 1: Estimates in the class Xδ,t. Let us take z0 ∈ Xδ,t. Then the solution of

(5.10) has constant norm since At is skew-adjoint (H1). Indeed,

zk+1 =(I + t

2 At

I − t2 At

)zk := Ttz

k,

where the operator Tt is obviously unitary.

Further, since

zk + zk+1

2=

1

2

(I + Tt

)zk =

( I

I − t2 At

)zk,

we get that for any k,

∥∥∥∥z0 + z1

2

∥∥∥∥2

X

=

∥∥∥∥zk + zk+1

2

∥∥∥∥2

X

≥ 1

1 +(

δ2

)2

∥∥z0∥∥2

X, (5.17)

as a consequence of (5.7) and the skew-adjointness assumption (H1) of At.

Step 2: The resolvent estimate. Set χ ∈ H1(lR) and χk = χ(kt). Let gk = χkzk,

and

fk =gk+1 − gk

t −A−t(gk+1 + gk

2

). (5.18)

One can easily check that

fk =χk+1 − χk

tzk+1 + zk

2+χk+1 + χk

2

zk+1 − zk

t

−At

(χk+1 + χk

2

zk+1 + zk

2+χk+1 − χk

2

zk+1 − zk

2

)

=χk+1 − χk

t(zk + zk+1

2− (t)2

4At

(zk+1 − zk

t))

=(χk+1 − χk

t)(I − (t)2

4A2

t

)(zk + zk+1

2

). (5.19)

Especially, recalling (5.17) and (5.7), (5.19) implies

∥∥∥fk∥∥∥

2

X≤(χk+1 − χk

t)2∥∥∥∥z0 + z1

2

∥∥∥∥2

X

(1 +

δ2

4

). (5.20)

In particular, fk ∈ l2(tZ;X).


Taking the Fourier transform of (5.18), for all τ ∈ (−π/t, π/t), we get

f(τ) = t∑

k∈Z

fk exp(−iktτ)

= t∑

k∈Z

(gk+1 − gk

t −At

(gk+1 + gk

2

))exp(−iktτ)

= t∑

k∈Z

(exp(itτ) − 1

t −At

(exp(itτ) + 1

2

))gk exp(−iktτ)

=(i

2

t tan(τt2

)I −At

)g(τ) exp

(iτt2

)cos(τt

2

).

(5.21)

We claim the following Lemma:

Lemma 5.2.1. The solution (zk) in (5.10) satisfies

(1 + α)m2t∑

k∈Z

(χk + χk+1

2

)2∥∥∥∥Bt

(zk + zk+1

2

)∥∥∥∥2

Y

≥∥∥∥∥z0 + z1

2

∥∥∥∥2

X

[a1t

∑

k∈Z

(χk + χk+1

2

)2− a2t

∑

k∈Z

(χk+1 − χk

t)2], (5.22)

with

a1 =(1 − 1

β

),

a2 = M2(1 +

δ2

4

)2+m2C2

B

(1 +

1

α

) δ416

+(t)2

4δ2(β − 1),

(5.23)

for any α > 0 and β > 1, where CB,M,m are as in (5.8)-(5.9).

Proof of Lemma 5.2.1. Let

G(τ) = g(τ) exp(iτt2

) cos(τt2

). (5.24)

By its definition and the fact that zk ∈ Xδ,t, it is obvious that G(τ) ∈ Xδ,t.

In view of (5.21), applying the resolvent estimate (5.9) to G(τ), integrating on τ from

−π/t to π/t, it holds

M2

∫ π/t

−π/t

∥∥∥f(τ)∥∥∥

2

Xdτ +m2

∫ π/t

−π/t‖BtG(τ)‖2

Y dτ

≥∫ π/t

−π/t‖G(τ)‖2

X dτ. (5.25)

Applying Parseval’s identity (5.16) to (5.25), and noticing that

Gk =gk + gk+1

2, i.e. G(τ) =

(gk + gk+1

2

)(τ),


we get

M2t∑

k∈Z

∥∥∥fk∥∥∥

2

X+m2t

∑

k∈Z

∥∥∥∥Bt

(gk + gk+1

2

)∥∥∥∥2

Y

≥ t∑

k∈Z

∥∥∥∥gk + gk+1

2

∥∥∥∥2

X

. (5.26)

Now we estimate the three terms in (5.26). The first term can be bounded above in

view of (5.20).

Second, since

gk+1 + gk

2=χk+1 + χk

2

zk+1 + zk

2+

t2

χk+1 − χk

tzk+1 − zk

2, (5.27)

using

‖a+ b‖2 ≤ (1 + α) ‖a‖2 +(1 +

1

α

)‖b‖2 ,

we deduce that

∥∥∥∥Bt

(gk+1 + gk

2

)∥∥∥∥2

Y

≤ (1 + α)(χk+1 + χk

2

)2∥∥∥∥Bt

(zk+1 + zk

2

)∥∥∥∥2

Y

+(1 +

1

α

) (t)416

(χk+1 − χk

t)2∥∥∥∥Bt

(zk+1 − zk

t)∥∥∥∥

2

Y

≤ (1 + α)(χk+1 + χk

2

)2∥∥∥∥Bt

(zk+1 + zk

2

)∥∥∥∥2

Y

+(1 +

1

α

) δ416C2

B

(χk+1 − χk

t)2∥∥∥∥z0 + z1

2

∥∥∥∥2

X

.

(5.28)

In (5.28) we use the fact that (recalling (5.8))

∥∥∥∥BtAt

(zk + zk+1

2

)∥∥∥∥Y

≤ CB

∥∥∥∥A2t

(zk + zk+1

2

)∥∥∥∥X

≤ δ2CB

(t)2∥∥∥∥z0 + z1

2

∥∥∥∥X

.

Finally, for any β > 1, recalling (5.17) and (5.27), we get

∥∥∥∥gk+1 + gk

2

∥∥∥∥2

X

≥(1 − 1

β

)(χk+1 + χk

2

)2∥∥∥∥zk+1 + zk

2

∥∥∥∥2

X

−(β − 1)(t

2

)2(χk+1 − χk

t)2∥∥∥∥zk+1 − zk

2

∥∥∥∥2

X

≥(1 − 1

β

) 1

1 + ( δ2 )2

(χk+1 + χk

2

)2 ∥∥z0∥∥2

X

−(β − 1)(t

2

)4(χk+1 − χk

t)2∥∥∥∥At

(z0 + z1

2

)∥∥∥∥2

X

,

(5.29)


where we used

‖a+ b‖2 ≥(1 − 1

β

)‖a‖2 −

(β − 1

)‖b‖2 .

Applying (5.20), (5.28) and (5.29) to (5.26), we complete the proof of Lemma 5.2.1. ♦

Step 3: The observability estimate. This step is aimed to derive the observability

estimate (5.11) stated in Theorem 5.2.2 from Lemma 5.2.1 with explicit estimates on the

optimal time Tδ.

First of all, let us recall the following classical Lemma on Riemann sums:

Lemma 5.2.2. Let χ(t) = φ(t/T ) with φ ∈ H2 ∩H10 (0, 1), extended by zero outside (0, T ).

Recalling that χk = χ(kt), the following estimates hold:

∣∣∣t∑

k∈Z

(χk + χk+1

2

)2− T ‖φ‖2

L2(0,1)

∣∣∣ ≤ 2Tt ‖φ‖L2(0,1)

∥∥∥φ∥∥∥

L2(0,1),

∣∣∣t∑

k∈Z

(χk+1 − χk

t)2

− 1

T

∥∥∥φ∥∥∥

2

L2(0,1)

∣∣∣ ≤ 2

Tt∥∥∥φ∥∥∥

L2(0,1)

∥∥∥φ∥∥∥

L2(0,1).

(5.30)

Sketch of the proof of Lemma 5.2.2. It is easy to show that for all f = f(t) ∈ C1(0, T )

and sequence τk ∈ [kt, (k + 1)t], it holds

∣∣∣∫ T

0f(t)dt−t

∑

k∈(0,T/t)

f(τk)∣∣∣ ≤

∑

k∈(0,T/t)

∫∫

[kt,(k+1)t]2|f(s)| ds dt

≤ t∫ T

0|f | dt. (5.31)

Replacing f by φ2 we get the first inequality (5.30). Similarly, replacing f by φ2, the second

one can be proved too. ♦

Taking Lemma 5.2.1 and 5.2.2 into account, the coefficient of∥∥(z0 + z1)/2

∥∥2

Xin (5.22)

tends to

kT,δ,α,β,φ =1

m2(1 + α)

[(1 − 1

β

)T ‖φ‖2

L2(0,1)

−(M2(1 +

δ2

4

)2+m2C2

B

(1 +

1

α

) δ416

) 1

T

∥∥∥φ∥∥∥

2

L2(0,1)

],

when t→ 0.

Note that kT,δ,α,β,φ is an increasing function of T tending to −∞ when T → 0+ and to

+∞ when T → ∞. Let Tδ,α,β,φ be the unique positive solution of kT,δ,α,β,φ = 0. Then, for

any time T > Tδ,α,β,φ, choosing a positive kT,δ such that

0 < kT,δ < kT,δ,α,β,φ,


there exists t0 > 0 such that for any t < t0, the following holds:

kT,δ

∥∥∥∥z0 + z1

2

∥∥∥∥2

X

≤ t∑

k∈(0,T/t)


2

Y. (5.32)

This combined with (5.17) yields (5.11).

This construction yields the following estimate on the time Tδ in Theorem 5.2.2. Namely,

for any α > 0, β > 1 and smooth function φ, compactly supported in [0, 1]:

Tδ ≤

∥∥∥φ∥∥∥

L2

‖φ‖L2

[ β

β − 1

]1/2[M2(1 +

δ2

4

)2+m2C2

B

(1 +

1

α

) δ416

]1/2.

We optimize in α, β and φ by choosing α = ∞, β = ∞ and

φ(t) =

sin(πt), t ∈ (0, 1)

0, elsewhere,(5.33)

which is well-known to minimize the ratio∥∥∥φ∥∥∥

L2

‖φ‖L2

.

For this choice of φ, this quotient equals π, and thus we recover the estimate (5.12). This

completes the proof of Theorem 5.2.2.♦

Theorem 5.2.1 has many applications. Indeed, it roughly says that, for any continuous

conservative system which is observable in finite time, there exists a time semi-discretization

which uniformly preserves the observability property in finite time, provided the initial data

are filtered at a scale 1/t. Later, using formally some microlocal tools, we will explain

why this filtering scale is the optimal one. Note that in Theorem 7.1 of [104] this scale was

proved to be optimal for a particular time-discretization scheme on the wave equation.

Besides, as we will see in Section 5.3, Theorem 5.2.2 is a key ingredient to address

observability issues.

5.3. General time-discrete schemes

5.3.1. General time-discrete schemes for first order systems

In this subsection, we consider a time semi-discrete scheme of (2.18)-(2.19) of the form

zk+1 = Ttzk, yk = Bzk, (5.34)

5.3. GENERAL TIME-DISCRETE SCHEMES 113

where Tt is a linear operator with the same eigenvectors as the operator A. We also

assume that the scheme is conservative in the sense that there exist real numbers λj,t such

that


Moreover, we assume that there is an explicit relation between λj,t and µj (as in (1.37))

of the following form:

λj,t =1

t h(µjt), (5.36)

where h : [−δ, δ] 7→ [−π, π] is a smooth strictly increasing function, i.e.

|h(η)| ≤ π, infh′(η), |η| ≤ δ > 0. (5.37)

Roughly speaking, the first part of (5.37) reflects the fact that one cannot measure frequen-

cies higher than π/t in a mesh of size t. Besides, the second part is a non-degeneracy

condition on the group velocity (see [94]) of solutions of (5.34) which is necessary to guar-

antee the propagation of solutions that is required for observability to hold.

We also assumeh(η)

η−→ 1 as η → 0. (5.38)

This guarantees the consistency of the time-discrete scheme with the continuous model

(2.18).

We have the following Theorem:


Under assumptions (5.35), (5.36), (5.37) and (1.52), for any δ > 0, there exists a time

Tδ such that for all T > Tδ, there exists a constant kT,δ > 0 such that for all t small

enough, any solution of (5.34) with initial value z0 ∈ Cδ/t satisfies

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥∥B(zk + zk+1

2

)∥∥∥∥2

Y

. (5.39)

Besides, we have the following estimate on Tδ:

Tδ ≤ π

[M2(1 + tan2

(h(δ)2

))2sup|η|≤δ

cos4(h(η)/2)

h′(η)2

+m2C2B sup

|η|≤δ

2

ηtan

(h(η)2

)2tan4

(h(δ)2

)]1/2

, (5.40)



Proof. The main idea is to use Theorem 5.2.2. Hence we introduce an operator At such

that the solution of (5.34) coincides with the solution of the linear system

zk+1 − zk

t = At

(zk + zk+1

2

), z0 = z0. (5.41)

This can be done defining the action of the operator At on each eigenfunction:

AtΨj = ikt(µj)Ψj , (5.42)

where

kt(ω) =2

t tan(h(ωt)

2

). (5.43)

Indeed, if

z0 =∑

ajΨj,

then the solution of (5.34) can be written as

zk =∑

ajφj exp(iλjkt) =∑

ajφj exp(ih(µjt)k)

and the definition of At follows naturally.

Obviously, when the scheme (5.34) under consideration is the one of Section 5.2, that is

(5.1), the operator At is precisely the operator A.

Then (5.39) would be a straightforward consequence of Theorem 5.2.2, if we could

prove the resolvent estimate for At. We will see in the sequel that a weak form of the

resolvent estimate holds, and that this is actually sufficient to get the desired observability

inequality. In the sequel, δ is a given positive number, determining the class of filtered data

under consideration.

Step 1: A weak form of the resolvent estimate. By hypothesis (2.22),

M2 ‖(A− iω)z‖2X +m2 ‖Bz‖2

Y ≥ ‖z‖2X , z ∈ D(A), ω ∈ lR. (5.44)

For z ∈ Cδ/t, that is

z =∑

|µj |≤δ/t

ajφj, (5.45)

one can easily check that

‖(A− iω)z‖2X =

∑|aj |2

(µj − ω

)2


and

‖(At − iω)z‖2X =

∑|aj|2

(kt(µj) − ω

)2.

Especially, for any ω ∈ lR, this last estimate takes the form

‖(At − ikt(ω))z‖2X =

∑|aj |2

(kt(µj) − kt(ω)

)2

with kt as in (5.43). Thus, taking ε > 0, it follows that for any ω < δ+εt ,

‖(At − ikt(ω))z‖2X ≥

(inf

|ω|t≤δ+ε

|k′t(ω)|

)2‖(A− iω)z‖2

X .

Hence, setting

αt,ε = kt

(δ + ε

t), Cδ,ε =

(infk′t(ω) : |ω|t ≤ δ + ε

)−1, (5.46)

we get the following weak resolvent estimate:

C2δ,εM

2∥∥∥(At − iω

)z∥∥∥

2

X+m2 ‖Bz‖2

Y ≥ ‖z‖2X , z ∈ Cδ/t, |ω| ≤ αt,ε. (5.47)

Our purpose is now to show that this is enough to get the time-discrete observability

estimate. We emphasize that the main difference between (5.47) and (5.9) is that in (5.9)

is assumed to hold for all ω ∈ lR while in (5.47) it is only assumed to hold for |ω| ≤ αt,ε.

Step 2: Improving the resolvent estimate (5.47). Here we prove that (5.47) can be

extended to all ω ∈ lR. Indeed, consider ω such that |ω| ≥ αt,ε and z ∈ Cδ/t as in (5.45).

Then

‖(At − iω)z‖2X ≥

∑

|µj |≤δ/t

(kt(µj) − kt

(δ + ε

t))2

a2j

≥∑

|µj |≤δ/t

(kt

( δ

t)− kt

(δ + ε

t))2

a2j

≥( ε

t)2(

infωt∈[δ,δ+ε]

k′t(ω))2

‖z‖2 .

Using the explicit expression (5.43) of kt, we get

‖(At − iω)z‖2X ≥

( ε

t)2

infη∈[δ,δ+ε]

h′(η)2 ‖z‖2 . (5.48)

Therefore, for each ε > 0, in view of (1.51), there exists (t)ε > 0 such that, for t ≤ (t)ε

C2δ,εM

2∥∥∥(At − iω

)z∥∥∥

2

X+m2 ‖Bz‖2

Y ≥ ‖z‖2X , z ∈ Cδ/t, ω ∈ lR. (5.49)


Step 3: Application of Theorem 5.2.2. First, one easily checks from (5.42)-(5.43)

that

t ‖Atz‖X ≤ δ ‖z‖X , z ∈ Cδ/t, (5.50)

with δ = 2 tan(h(δ)/2

).

Second, we check that there exist a constant CB,δ such that

‖Bz‖Y ≤ CB,δ ‖Atz‖X , z ∈ Cδ/t, (5.51)

where CB is as in (5.4). Indeed, for z ∈ Cδ/t,

‖Az‖X ≤ sup|ω|t≤δ

∣∣∣kt(ω)

ω

∣∣∣‖Atz‖X ,

and therefore one can take

CB,δ = βδCB , (5.52)

where

βδ = sup|η|≤δ

2

ηtan

(h(η)2

),

which is finite from hypothesis (5.37) and (1.52).

Third, the resolvent estimate (5.49) holds.

Then Theorem 5.2.2 can be applied and proves the observability inequality (5.39) for

the solutions of (5.34) with initial data in Cδ/t. Besides, we have the following estimate

on the observability time Tδ,ε :

Tδ,ε = π[(

1 +δ2

4

)2M2C2

δ,ε +m2C2Bβ

2δ

δ4

16

]1/2.

In the limit ε → 0, Tδ,ε converges to an observability time Tδ. Besides, using the explicit

form of the constants Cδ,ε, δ and βδ one gets (5.40). ♦

5.3.2. The Newmark method for second order in time systems

In this subsection we investigate observability properties for time-discrete schemes of

the second order in time evolution equation.

More precisely, let H be a Hilbert space endowed with the norm ‖·‖H and let A0 :

D(A0) → H be a self-adjoint positive operator with compact resolvent. We consider the

second order in time system

utt(t) +A0u(t) = 0,

u(0) = u0, ut(0) = v0,(5.53)


where A0 is a positive operator. Such systems can be seen as a generic model for the

free vibrations of elastic structures such as strings, beams, membranes, plates or three-

dimensional elastic bodies.


E(t) = ‖ut(t)‖2H +

∥∥∥A1/20 u(t)

∥∥∥2

H, (5.54)


We consider the output function

y(t) = B1u(t) +B2ut(t), (5.55)

where B1 and B2 are two observation operators satisfying B1 ∈ L(D(A0), Y ) and B2 ∈L(D(A

1/20 ), Y ). In other words, we assume that there exists two constants CB,1 and CB,2,

such that

‖B1u‖Y ≤ CB,1 ‖A0u‖H , ‖B2v‖Y ≤ CB,2

∥∥∥A1/20 v

∥∥∥ . (5.56)

In the sequel, we assume either B1 = 0 or B2 = 0. Indeed, it is a necessary condition on

the time-discrete level, due to a technical problem that we cannot solve so far, as we shall

see in the proof of Theorem 5.3.2.

System (5.53)–(5.55) can be put in the form (2.18)–(2.19). Indeed, setting

z1(t) = ut + iA1/20 u, z2(t) = ut − iA

1/20 u, (5.57)

equation (5.53) is equivalent to

zt = Az, z =

(z1z2

), A =

(iA

1/20 0

0 −iA1/20

), (5.58)

for which the energy space is X = H × H with the domain D(A) = D(A1/20 ) × D(A

1/20 ).

Moreover, the energy E(t) given in (5.54) coincides with half of the norm of z in X. Note

that the spectrum of A is explicitly given by the spectrum of A0. Indeed, if (µ2j )j∈N∗ (µj > 0)

is the sequence of eigenvalues of A0, i.e.


∗,

with corresponding eigenvectors φj , then the eigenvalues of A are ±iµj, with corresponding

eigenvectors

Ψj =

(φj

0

), Ψ−j =

(0

φj

), j ∈ lN∗. (5.59)


Besides, in the new variables (5.57), the output function is given by

y(t) = Bz(t) = B1A−1/20

( iz2(t) − iz1(t)

2

)+B2

(z1(t) + z2(t)

2

). (5.60)

Recalling the assumptions on B1 and B2 in (5.56), the admissible observation B belongs to

L(D(A), Y ).

In the sequel, we assume that the system (5.53)–(5.55) is exactly observable. As a direct

consequence of this we obtain that system (5.58)–(5.60) is exactly observable and therefore

the resolvent estimate (2.22) holds.

We now introduce the time-discrete schemes we are interested in. For any t > 0 and

β > 0, we consider the following Newmark time-discrete schemes for the system (5.53):

uk+1 + uk−1 − 2uk

(t)2 +A0

(βuk+1 + (1 − 2β)uk + βuk−1

)= 0,

(u0 + u1

2,u1 − u0

t)

= (u0, v0) ∈ D(A120 ) ×H.

(5.61)


Ek+1/2 =

∥∥∥∥A1/20

(uk + uk+1

2

)∥∥∥∥2

+

∥∥∥∥uk+1 − uk

t

∥∥∥∥2

+ (4β − 1)(t)2

4

∥∥∥∥A1/20

(uk+1 − uk

t)∥∥∥∥

2

, k ∈ Z, (5.62)

which is a discrete counterpart of the continuous energy (5.54). Multiplying the first equa-

tion of (5.61) by (uk+1 − uk−1)/2t, taking its norm and using integration by parts, it is

easy to show that (5.62) remains constant with respect to k. Furthermore, we assume in

the sequel that β ≥ 1/4 to guarantee that system (5.61) is unconditionally stable.

The output function is given by the following discretization of (5.55):

yk+1/2 = B1

(uk + uk+1

2

)+B2

(uk+1 − uk

t), (5.63)

where, as in (5.55), we assume that either B1 or B2 vanishes.

As before, for any s > 0, we define (as in (1.38)),

Cs = span Ψj : the corresponding iµj satisfies |µj| ≤ s. (5.64)

Note that this space is invariant under the actions of the discrete semi-groups associated to

the Newmark time-discrete schemes (5.61).

We have the following theorem:


Theorem 5.3.2. Let β ≥ 1/4 and δ > 0. We assume that either B1 ≡ 0 or B2 ≡ 0.

Then there exists a time Tδ such that for all T > Tδ, there exists a positive constant

kT,δ, such that for t small enough, the solution of (5.61) with initial data (u0, v0) ∈ Cδ/t

satisfies

kT,δE1/2 ≤ t

∑

kt∈(0,T )

∥∥∥yk+1/2∥∥∥

2

Y, (5.65)

where yk+1/2 is defined in (5.63) and B1, B2 satisfy (5.56).

Besides, Tδ can be chosen as

Tδ,1 = π[(1 + βδ2)2

(1 +

(β − 1

4)δ2)2M2 +m2C2

B,1

δ

16

4]1/2, (5.66)

if B2 = 0 and as

Tδ,2 = π[(1 + βδ2)2

(1 +

(β − 1

4

)δ2)M2 +m2C2

B,2

δ4

16

]1/2, (5.67)

if B1 = 0.

Remark 5.3.1. This result, and especially the time estimates (5.66) and (5.67) on the

observability times need further comments.

As in Theorem 5.2.1, we see that, if we filter at a scale smaller than t, for instance in

the class Cδ/(t)α, with α < 1, then the uniform observability time T0 is given by T0 = πM ,

which coincides with the estimate obtained by the resolvent estimate (2.22) in the continuous

setting.

Note that the estimates (5.66) and (5.67) do not have the same growth in δ when δ goes

to ∞. This fact does not seem to be natural because the observability time is expected to

depend on the group velocity (see [94]) and not to the form of the observable.

By now we could not avoid the assumption that either B1 or B2 vanishes, the special

case β = 1/4 being excepted.

However, we can deal with an observable of the form

yk+1/2 = B1

√I + (β − 1/4)(t)2A0

(uk + uk+1

2

)+B2

(uk+1 − uk

t), (5.68)

with both non-trivial B1 and B2. Indeed, in this case, the operator Bt arising in the proof

of Theorem 5.3.2 does not depend on t and therefore the proof works as in the case B1 = 0,

with the time estimate (5.67). However, this observation operator, which compares to the

continuous one (5.55) when δ → 0, does not seem to be the most natural discretization of

(5.60).


When β = 1/4, both (5.66) and (5.67) have the same form. Besides, one can easily

adapt the proof to show that when β = 1/4, we can deal with a general observation operator

B as in (5.55). Actually, the Newmark scheme (5.61) with β = 1/4 is equivalent to a

midpoint scheme, and therefore Theorem 5.2.1 applies.

Proof. Step 1. We first transform system (5.61) into a first order time-discrete scheme

similar to (5.58). For this, we define

A0,t = A0[I + (β − 1/4)(t)2A0]−1. (5.69)

Then (5.61) can be rewritten as

uk+1 + uk−1 − 2uk

(t)2 +A0,t

(uk−1 + 2uk + uk+1

4

)= 0. (5.70)

As in (5.57), using the following change of variables

zk+1/21 =

uk+1 − uk

t + iA1/20,t

(uk + uk+1

2

),

zk+1/22 =

uk+1 − uk

t − iA1/20,t

(uk + uk+1

2

),

(5.71)

system (5.61) (and also system (5.70)) is equivalent to

zk+1/2 − zk−1/2

t = At

(zk−1/2 + zk+1/2

2

), (5.72)

with

At =

(iA

1/20,t 0

0 −iA1/20,t

), zk+1/2 =

z

k+1/21

zk+1/22

. (5.73)

Consequently, the observation operator yk+1/2 (5.63) is given by

yk+1/2 = B1A−1/20,t

( izk+1/22 − iz

k+1/21

2

)+B2

(zk+1/21 + z

k+1/22

2

)(5.74)

= Btz

k+1/2.

Step 2. We now verify that system (5.72)–(5.74) satisfies the hypothesis of Theorem

5.2.2.

We first check (H1). It is obvious that the eigenvectors of At are the same as those of

A (see (5.59)). Moreover, for any Ψj we compute

AtΨj = ℓjΨj, with ℓj =iµj√

1 + (β − 1/4)(t)2µ2j

. (5.75)


In other words, we are close to the situation considered in Subsection 5.3.1, and the time

semi-discrete approximation scheme (5.72) satisfies the hypothesis (5.35), (5.36), (5.37),

(1.51) and (1.52) with the function h defined by

h(η) = 2 arctan(η

2

1√1 + (β − 1/4)η2

)(5.76)

In particular, this implies that (5.50) holds in the class Cδ/t, and takes the form

t ‖Atz‖X ≤ δ√1 + (β − 1/4)δ2

‖z‖X , z ∈ Cδ/t. (5.77)

Second, we check hypothesis (H2).

‖Btz‖Y ≤ ‖Atz‖H

(CB,1

∥∥∥A0A−10,t

∥∥∥L(Cδ/t,H)

+CB,2

∥∥∥A1/20 A

−1/20,t

∥∥∥L(Cδ/t,H)

)

≤ ‖Atz‖H

((1 + (β − 1/4)δ2)CB,1 +

√1 + (β − 1/4)δ2CB,2

)

≤ CB,δ ‖Atz‖H . (5.78)

The third point is more technical. Following the proof of Theorem 5.3.1, for any ε > 0,

we obtain the following resolvent estimate:

C2δ,εM

2∥∥∥(At − iω

)z∥∥∥

2

X+m2 ‖Bz‖2

Y ≥ ‖z‖2X , z ∈ Cδ/t, ω ∈ lR, (5.79)

where Cδ,ε is given by (5.46), with

kt(ω) =ω√

1 + (β − 1/4)(ωt)2.

Straightforward computations show that actually

Cδ,ε =(1 + (β − 1/4)(δ + ε)2

)3/2. (5.80)

Our goal now is to derive from (5.79) the resolvent estimate (H3) given in (5.9). Here, we

will handle separately the two cases B1 = 0 and B2 = 0.

The case B1 = 0. Under this assumption, Bt = B, and therefore, (5.79) is the resolvent

estimate (H3) we need.

The case B2 = 0. In this case, we remark the following factorization

Btz = BRtz, where Rt =

(A

1/20 A

−1/20,t 0

0 A1/20 A

−1/20,t

)= AA−1

t.


Note that the operator Rt commutes with At, maps Cδ/t into itself, and is invertible.

Then, applying (5.79) to Rtz, we obtain that

C2δ,εM

2∥∥∥Rt

(At − iω

)z∥∥∥

2

X+m2 ‖Btz‖2

Y ≥ ‖Rtz‖2X ,

z ∈ Cδ/t, ω ∈ lR. (5.81)

We now compute explicitly the norm of Rt and R−1t in the class Cδ/t. Since

A0A−10,t = 1 + (β − 1/4)(t)2A0,

one easily checks that

‖Rt‖2δ = 1 + (β − 1/4)δ2,

∥∥∥R−1t

∥∥∥2

δ= 1, (5.82)

where ‖·‖δ denotes the operator norm from Cδ/t into itself. Applying (5.82) into (5.81),

we obtain

C2δ,εM

2(1 + (β − 1/4)δ2

) ∥∥∥(At − iω

)z∥∥∥

2

X+m2 ‖Btz‖2

Y ≥ ‖z‖2X ,

z ∈ Cδ/t, ω ∈ lR. (5.83)

Thus, in both cases, we can apply Theorem 5.2.2, which gives the existence of a time

Tδ,ε such that for T > Tδ,ε, there exist a positive kT,δ such that

kT,δ

∥∥∥z1/2∥∥∥

2

X≤

T/t∑

k=0

∥∥∥Btzk+1/2

∥∥∥2

Y.

Besides, the estimates of Theorem 5.2.2 allows to estimate the time Tδ,ε

Tδ,ε =

π[(1 + βδ2)2

(1 + (β − 1/4)(δ + ε)2)3

1 + (β − 1/4)δ2M2 +m2C2

B,1

δ

16

4]1/2,

if B2 = 0,

π[(1 + βδ2)2

(1 + (β − 1/4)(δ + ε)2)3

(1 + (β − 1/4)δ2)2M2 +m2C2

B,2

δ4

16

]1/2,

if B1 = 0.

Letting ε→ 0, we obtain the estimates (5.66)-(5.67).

To complete the proof we check that if the initial data z1/2 is taken within the class

Cδ/t, the solution of (5.61) satisfies

∥∥∥z1/2∥∥∥

2

X=∥∥∥zk+1/2

∥∥∥2

X≥ 2

1 + (β − 1/4)δ2Ek+1/2,

which can be deduced by the explicit expression of the solution of (5.61) and the formula

(5.71). ♦

5.4. APPLICATIONS 123

5.4. Applications

5.4.1. Application of Theorem 5.2.1

Boundary observation of the Schrodinger equation

The goal of this subsection is to present a straightforward application of Theorem 5.2.1

to the observability properties of the Schrodinger equation based on the results in [48].

Let Ω ⊂ lRn be a bounded domain. Consider the equation

iut = ∆u, (t, x) ∈ (0, T ) × Ω,

u(0) = u0, x ∈ Ω, u(t, x) = 0, (t, x) ∈ (0, T ) × ∂Ω.(5.84)

where u0 ∈ L2(Ω) is the initial data. Equation (5.84) can obviously be put under the form

(2.18) by setting A = −i∆, whose domain is D(A) = H10 (Ω) ∩H2(Ω).

Let Γ0 ⊂ ∂Ω be an open subset of ∂Ω and define the output

y(t) =∂u(t)

∂ν

∣∣∣Γ0

.

Using Sobolev’s embedding theorems, one can easily check that this defines a continuous

observation operator B from D(A) to L2(Γ0).

Let us assume that Γ0 satisfies in some time T the Geometric Control Condition (GCC)

introduced in [2], which asserts that all rays of Geometric optics in Ω touch the sub-boundary

Γ0 in a time smaller than T . In this case, the following observability result is known ([48]) :

Theorem 5.4.1. For any T > 0, there exist positive constants kT > 0 and KT > 0 such

that for any u0 ∈ L2(Ω), the solution of (5.84) satisfies

kT ‖u0‖2L2(Ω) ≤

∫ T

0

∫

Γ0

∣∣∣∂u(t)

∂ν

∣∣∣2dΓ0dt ≤ KT ‖u0‖2

L2(Ω) . (5.85)

We then introduce the following time semi-discretization of system (5.84):

iuk+1 − uk

t = ∆(uk+1 + uk

2

), x ∈ Ω, k ∈ N

uk(x) = 0, x ∈ ∂Ω, k ∈ N

u0(x) = u0(x), x ∈ Ω,

(5.86)

that we observe through

yk =∂uk

∂ν

∣∣∣Γ0

.

Theorem 5.2.1 implies the following result:


Theorem 5.4.2. For any δ > 0, there exists a time Tδ such that for any time T > Tδ, there

exists a positive constant kT,δ > 0 such that for t small enough, the solution of (5.86)

satisfies

kT ‖u0‖2L2(Ω) ≤ t

∑

k∈(0,T/t)

∫

Γ0

∣∣∣∂uk

∂ν

∣∣∣2dΓ0 (5.87)

for any u0 ∈ Cδ/t.

Remark 5.4.1. Note that in the present section, we do not state any admissibility result for

the time-discrete systems under consideration. However, uniform (with respect to t > 0)

admissibility results hold for all the examples presented in this Chapter. These results will

be derived in Section 5.6 using the admissibility property of the continuous system (2.18)-

(2.19).

Boundary observation of the linearized KdV equation

We now present an application of Theorem 5.2.1 to the boundary observability of the

linear KdV equation. We consider the following initial-value boundary problem for the KdV

equation:

ut + uxxx = 0, (t, x) ∈ (0, T ) × (0, 2π),

u(t, 0) = u(t, 2π), t ∈ (0, T ),

ux(t, 0) = ux(t, 2π), t ∈ (0, T ),

uxx(t, 0) = uxx(t, 2π), t ∈ (0, T ),

u(0, x) = u0(x), x ∈ (0, 2π).

(5.88)

For any integer k we set

Hkp

=u ∈ Hk(0, 2π); ∂j

xu(0) = ∂jxu(2π) for 0 ≤ j ≤ k − 1

, (5.89)

where Hk(0, 2π) denotes the classical Sobolev spaces on the interval (0, 2π). The initial data

of (5.88) are taken in the space X= H2

p (0, 2π), endowed with the classical H2(0, 2π)-norm.

Let A denote the operator Au = −∂3xu of domain D(A) = H5

p . As it is mentioned in

[88], A is a skew-adjoint operator with compact resolvent. Moreover, its spectrum is given

by σ(A) = iµj with µj = j3, j ∈ Z. The output function y(t) and the corresponding

operator B : D(A) −→ Y is given by

y(t)= Bu(t) =

u(t, 0)

ux(t, 0)

uxx(t, 0)

,


with the norm ‖Bu‖2Y = |u(0)|2 + |ux(0)|2 + |uxx(0)|2. Note that B ∈ L(H5

p , lR3).

The following observability inequality for system (5.88) is well-known (Prop. 2.2 of [85]):

Lemma 5.4.1. Let T > 0. Then there exist positive numbers kT and KT such that for

every u0 ∈ H2p(0, 2π),

kT ‖u0‖2H2

p≤∫ T

0

(|u(t, 0)|2 + |ux(t, 0)|2 + |uxx(t, 0)|2

)dt ≤ KT ‖u0‖2

H2p. (5.90)

We now introduce the following time semi-discretization of system (5.88):

uk+1 − uk

t +uk+1

xxx + ukxxx

2= 0, x ∈ (0, 2π), k ∈ N

uk(0) = uk(2π), k ∈ N

ukx(0) = uk

x(2π), k ∈ N

ukxx(0) = uk

xx(2π), k ∈ N

u0(x) = u0(x), x ∈ (0, 2π).

(5.91)

It is easy to show that the eigenfunctions of A are given by Ψj = eijxj∈Z with the

corresponding eigenvalues ij3j∈Z. Hence, for any δ > 0, we have

Cδ/t = span Ψj , j3 ≤ δ/t. (5.92)

As a direct consequence of Theorem 5.2.1 we have the following uniform observability result

for system (5.91):

Theorem 5.4.3. For any δ > 0, there exists a time Tδ such that for any T > Tδ, there

exists a positive constant kT,δ > 0 such that for t small enough, the solution uk of (5.91)

satisfies

kT,δ ‖u0‖2H2

p≤ t

∑

kt∈(0,T )

(|uk(0)|2 + |uk

x(0)|2 + |ukxx(0)|2

), (5.93)

for any initial data u0 ∈ Cδ/t.


Let us present an application of Theorem 5.3.1 to the so-called fourth order Gauss

method discretization of equation (2.18) (see for instance [27]-[28]). This fourth order

Gauss method is a special case of the Runge-Kutta time approximation schemes, which

corresponds to the only conservative scheme within this class.


Consider the following discrete system:

κi = A(zk + t

2∑

j=1

αijκi

), i = 1, 2,

zk+1 = zk +t2

(κ1 + κ2),

z0 ∈ Cδ/t given,(αij) =

( 14

14 −

√3

614 +

√3

614

).

(5.94)

The scheme is unstable for the eigenfunctions corresponding to the eigenvalues µj such

that µjt ≥ 2√

3 ([27]-[28]). Thus we immediately impose the following restriction on the

filtering parameter :

δ < 2√

3.

To use Theorem 5.3.1, we only need to check the behavior of the semi-discrete scheme

(5.94)E on the eigenvectors. If z0 = Ψj, an easy computation shows that

z1 = exp(iℓjt)z0,

where

ℓj =2

t arctan( µjt

2 − (µjt)2/6). (5.95)

In other words, ℓjt = h(µjt), where

h(η) = 2 arctan( η

2 − η2/6

).

Then, a simple application of Theorem 5.3.1 gives :

Theorem 5.4.4. Assume that B is an observation operator such that (A,B) satisfy (2.22)

and B ∈ L(D(A), Y ).

For any δ ∈ (0, 2√

3), there exists a time Tδ > 0 such that for any T > Tδ, there exists

t0 such that for all 0 < t < t0, there exists a constant kT,δ > 0, independent of t,such that the solutions of the system (5.94) satisfy

kT,δ

∥∥z0∥∥2

X≤ t

∑

k∈(0,T/t)

∥∥∥Bzk∥∥∥

2

Y, ∀ z0 ∈ Cδ/t. (5.96)

Note that Theorem 5.3.1 also provides an estimate on Tδ by using (5.40).

In particular, this provides another possible time-discretization of (5.88), for which the

observability inequality holds uniformly in t provided the initial is taken in Cδ/t, with

δ < 2√

3, where Cδ/t is as in (5.92).



There are plenty of applications of Theorem 5.3.2. We present here an application to

the boundary observability of the wave equation.

Consider a smooth nonempty open bounded domain Ω ⊂ lRd and let Γ0 be an open

subset of ∂Ω. We consider the following initial boundary value problem:

utt − ∆u = 0, x ∈ Ω, t ≥ 0,

u(x, t) = 0, x ∈ ∂Ω, t ≥ 0,

u(x, 0) = u0, ut(x, 0) = v0, x ∈ Ω

(5.97)

with the output

y(t) =∂u

∂ν

∣∣∣Γ0

. (5.98)

This system is conservative and the energy of (5.97)

E(t) =1

2

∫

Ω

[|ut(t, x)|2 + |∇u(t, x)|2

]dx, (5.99)

remains constant, i.e.

E(t) = E(0), ∀ t ∈ [0, T ]. (5.100)

The boundary observability for the system is described as: For some constant C =

C(T,Ω,Γ0) > 0, solutions of (5.97) satisfy

E(0) ≤ C

∫ T

0

∫

Γ0

∣∣∣∂u

∂ν

∣∣∣2dΓ0dt, ∀ (u0, v0) ∈ H1

0 (Ω) × L2(Ω). (5.101)

Note that this inequality holds true for all triples (T,Ω,Γ0) satisfying the Geometric Control

Condition (GCC) introduced in [1], which asserts that all rays of Geometric Optics in Ω

touch the sub-boundary Γ0 in a time smaller than T . In this case, (5.101) is established by

means of micro-local analysis tools (see [1]). From now, we assume this condition to hold.

We then introduce the following time semi-discretization of (5.97):

uk+1 + uk−1 − 2uk

(t)2 =∆(βuk+1 + (1 − 2β)uk + βuk−1

), in Ω × Z,

uk = 0, in ∂Ω × Z,(u0 + u1

2,u1 − u0

t)

= (u0; v0) ∈ H10 (Ω) × L2(Ω),

(5.102)

where β is a given parameter satisfying β ≥ 14 .

The output functions yk are given by

yk =∂uk

∂ν

∣∣∣Γ0

. (5.103)


System (5.97)–(5.98) (or system (5.102)–(5.103)) can be written in form (5.53) (or (5.61))

with observation operator (5.55) if we introduce the following notations:

H = L2(Ω), D(A0) = H2(Ω) ∩H10 (Ω), Y = L2(Γ0),

A0ϕ = −∆ϕ ∀ϕ ∈ D(A0),

B1ϕ =∂ϕ

∂ν

∣∣∣Γ0

, ϕ ∈ D(A0).

One can easily check that A0 is self-adjoint in H, positive and boundedly invertible and

D(A1/20 ) = H1

0 (Ω), D(A1/20 )∗ = H−1(Ω).

Proposition 5.4.1. With the above notation, B1 ∈ L(D(A0), Y ) is an admissible observa-

tion operator, i.e. for all T > 0 there exists a constant KT > 0, independent of t, such

that: If u satisfies (5.97) then

∫ T

0

∫

Γ0

∣∣∣∂u

∂ν

∣∣∣2dΓ0dt ≤ KT

(‖u0‖2

H10 (Ω) + ‖v0‖2

L2(Ω)

)

for all (u0, v0) ∈ H10 (Ω) × L2(Ω).

The above proposition is classical (see, for instance, p. 44 of [56]), so we skip the proof.

Hence we are in the position to give the following theorem:

Theorem 5.4.5. Set β ≥ 1/4.

For any δ > 0, system (5.102) is uniformly observable with (u0, v0) ∈ Cδ/t. More

precisely, there exists Tδ, such that for any T > Tδ, there exists a positive constant kT,δ

independent of t, such that the solutions of system (5.102) satisfy

kT,δ

(‖u0‖2 + ‖v0‖2

)≤ t

∑

k∈(0,T/t)

∫

Γ0

∣∣∣∂uk

∂ν

∣∣∣2dΓ0, (5.104)

for any (u0, v0) ∈ Cδ/t.

Proof. The scheme proposed here comes from a Newmark discretization. Hence this result

is a direct consequence of Theorem 5.3.1. ♦

Remark 5.4.2. One can use Fourier analysis and microlocal tools to discuss the optimality

of the filtering condition as in [104]. The symbol of the operator in (5.102), that can be

obtained by taking the Fourier transform of the differential operator in space-time is of the

form ( see for instance [62])

4

t2 sin2(τt

2

)−∣∣∣ξ∣∣∣2(

1 − 4β sin2(τt

2

)).

5.5. FULLY DISCRETE SCHEMES 129

In this case, one expects the optimal observability time to be the time needed by all the

rays to meet Γ0 as in the continuous case. Along the bicharacteristics derived from this

hamiltonian the following identity holds

|τ | =2

t arctan

(|ξ|t

2

1√1 + (β − 1/4)|ξ|2(t)2

).

These rays are straight lines as in the continuous case, but their velocity is not 1 anymore.

Indeed, one can prove that along the rays corresponding to |ξ| < δ/t, the velocity of

propagation is given by∣∣∣dx

dt

∣∣∣ =1

1 + β(|ξ|t)21√

1 + (β − 1/4)(ξt)2≥ 1

(1 + βδ2)√

1 + (β − 1/4)δ2.

In other words, in the class Cδ/t, the velocity of propagation of the rays concentrated in

frequency around δ/t is (1+δ2/4)−1 times that of the continuous wave equation. Therefore

we expect that the optimal observability time T ∗δ in the class Cδ/t is

T ∗δ = T ∗

0 (1 + βδ2)

√1 +

(β − 1

4

)δ2, (5.105)

where T ∗0 is the optimal observability time for the continuous system. According to this,

the estimate Tδ,2 (5.67) on the time of observability has the good growth rate when δ → ∞.

Besides, when δ goes to ∞, we have that

Tδ,2 ≃ πM(1 + βδ2)

√1 +

(β − 1

4

)δ2. (5.106)

Recall that πM = T0 is the time of observability that the resolvent estimate (2.22) yields

in the continuous setting (see Remark 5.3.1). The similarity between (5.105) and (5.106)

indicates that the resolvent method accurately measures the group velocity.

Note however that T0 = πM is not the expected sharp constant (5.105). But this is one

of the drawbacks of the method based on the resolvent estimates we use in this Chapter.

Even at the continuous level the observability time one gets this way is far from being the

optimal one that Geometric Optics yields.

5.5. Fully discrete schemes

5.5.1. Main statement

In this section, we deal with the observability properties for time-discretization systems

such as (2.18)-(2.19) depending on an extra parameter, for instance the space mesh-size, or

the size of the microstructure in homogenization.


To this end, it is convenient to introduce the following class of operators:

Definition 5.5.1. For any (m,M,CB) ∈ (lR∗+)3, we define C(m,M,CB) as the class of

operators (A,B) satisfying:

(A1) The operator A is skew-adjoint on some Hilbert space X, and has a compact resolvent.

(A2) The operator B is defined from D(A) with values in a Hilbert space Y , and satisfies

(5.4) with CB.

(A3) The pair of operators (A,B) satisfies the resolvent estimate (2.22) with constants m

and M .

In this class, Theorems 5.2.1-5.3.1-5.3.2 apply and provide uniform observability results

for any of the time semi-discrete approximations schemes (5.1)-(5.2), (5.34), and (5.53).

Indeed, this can be deduced by the explicit form of the constants Tδ and kT,δ which only

depend on m,M and CB . Note that this definition does not depend on the spaces X and

Y .

When considering families of pairs of operators (A,B), it is not easy, in general, to show

that they belong to the same class C(m,M,CB) for some choice of the constants (m,M,CB).

Indeed, item (A3) is not obvious in general. Therefore, in the sequel, we define another

class included in some C(m,M,CB) and that is easier to handle in practice.

Definition 5.5.2. For any (CB , T, kT ,KT ) ∈ (lR∗+)4, we define D(CB , T, kT ,KT ) as the

class of operators (A,B) satisfying (A1), (A2) and:

(B1) The admissibility inequality

∫ T

0

∥∥B exp(tA)z0∥∥2

Ydt ≤ KT

∥∥z0∥∥2

X, (5.107)

where exp(tA) stands for the semigroup associated to the equation

zt = Az, z(0) = z0 ∈ X. (5.108)

(B2) The observability inequality

kT

∥∥z0∥∥2

X≤∫ T

0

∥∥B exp(tA)z0∥∥2

Ydt. (5.109)

As we will see below, assumptions (B1)-(B2) imply (A3):


Lemma 5.5.1. If the pair (A,B) belongs to D(CB , T, kT ,Kt), then there exist m and M

such that (A,B) ∈ C(m,M,CB).

Besides m and M can be chosen as

m =

√2T

kT, M = T

√KT

2kT. (5.110)

Proof. We only need to prove (A3). This is actually already done in [70] or in [95]. Indeed,

it was proved that once the admissibility inequality (2.20) and the observability inequality

(2.21) hold for some time T , then the resolvent estimate (2.22) hold with m and M as in

(5.110).♦

Note that assumptions (B1)-(B2) are related to the continuous systems (5.108).

Now we consider a sequence of operators (Ap, Bp) depending on a parameter p ∈ P ,

which are in some L(Xp) × L(D(Ap), Yp) for each p, where Xp and Yp are Hilbert spaces.

We want to address the observability problem for a time-discretization scheme of

zt = Apz, z(0) = z0 ∈ Xp, y(t) = Bpz(t) ∈ Yp. (5.111)

In applications, we need the observability to be uniform in both p ∈ P and t > 0 small

enough. The previous analysis and the properties of the class D(CB , T, kT ,KT ) suggest the

following two-steps strategy:

1. Study the continuous system (5.111) for every parameter p and prove the uniform

admissibility (5.107) and observability (5.109).

2. Apply one of the Theorems 5.2.1, 5.3.1 and 5.3.2 to obtain uniform observability

estimates (5.3) for the corresponding time-discrete approximation schemes.

This allows dealing with fully discrete approximation schemes. In that setting the

parameter p is actually the standard parameter h > 0 associated with the space mesh-size.

In this way one can use automatically the existing results for space semi-discretizations as,

for instance, [5], [12], [13], [32], [75], [74], [108] and [110], etc.

5.5.2. Applications

The fully discrete wave equation

Let us consider the wave equation (5.97) in a 2-d square. More precisely, let Ω =

(0, π)× (0, π) ⊂ lR2 and Γ0 be a subset of the boundary of Ω constituted by two consecutive


sides, for instance,

Γ0 = (x1, π) : x1 ∈ (0, π) ∪ (π, x2) : x2 ∈ (0, π) = Γ1 ∪ Γ2.

As in (5.98), the output function y(t) = Bu(t) is given by

Bu =∂u

∂ν

∣∣∣Γ0

=∂

∂x2u(x1, π)

∣∣∣Γ1

+∂

∂x1u(π, x2)

∣∣∣Γ2

.

Let us first consider the finite-difference semi-discretization of (5.97). The following can

be found in [108]. Given J,K ∈ lN we set

h1 =π

J + 1, h2 =

π

K + 1. (5.112)

We denote by ujk(t) the approximation of the solution u of (5.97) at the point xjk =

(jh1, kh2). The space semi-discrete approximation scheme of (5.97) is as follows:

utt,jk − uj+1k + uj−1k − 2ujk

h21

− ujk+1 + ujk−1 − 2ujk

h22

= 0

0 < t < T, j = 1, · · · , J ; k = 1, · · · ,Kujk = 0, 0 < t < T, j = 0, J + 1; k = 0,K + 1

ujk(0) = ujk,0, ujk(t, 0) = ujk,1, j = 1, · · · , J ; k = 1, · · · ,K.

(5.113)

System (5.113) is a system of JK linear differential equations. Moreover, if we denote

the unknown

U(t) = (u11(t), u21(t), · · · , uJ1(t), · · · , u1K(t), u2K(t), · · · , uJK(t))T ,

then system (5.113) can be rewritten in vector form as follows

Utt(t) +AhU(t) = 0, 0 < t < T.

U(0) = Uh,0, Ut(0) = Uh,1,(5.114)

where (Uh,0, Uh,1) = (ujk,0, ujk,1)1≤j≤J,1≤k≤K ∈ lR2JK are the initial data. The correspond-

ing solution of (5.113) is given by (Uh, Ut,h) = (ujk, u1,jk)1≤j≤J,1≤k≤K . Note that the entries

of Ah belonging to MJK(lR) may be easily deduced from (5.113).

As a discretization of the output, we choose

(BhU)j =ujK

h2, j = 1 · · · J, (5.115)

(BhU)J+k =uJk

h1, k = 1, · · ·K, (5.116)


The corresponding norm for the observation operator Bh is given by

‖BhU(t)‖2Yh

= h1

J∑

j=1

∣∣∣ujK(t)

h2

∣∣∣2+ h2

K∑

k=1

∣∣∣uJk(t)

h1

∣∣∣2.

Besides, the energy of the system (5.114) is given by

Eh(t) =h1h2

2

J∑

j=0

K∑

k=0

(|u1,jk(t)|2+

∣∣∣uj+1k(t) − ujk(t)

h1

∣∣∣2+∣∣∣ujk+1(t) − ujk(t)

h2

∣∣∣2). (5.117)

As in the continuous case, this quantity is constant.

Eh(t) = Eh(0), ∀ 0 < t < T.

In order to prove the uniform observability of (5.114), we have to filter the high frequencies.

To do that we consider the eigenvalue problem associated with (5.114):

Ahϕ = λ2ϕ. (5.118)

As in the continuous case, it is easy to show that the eigenvalues λj,k,h1,h2 are purely

imaginary. Let us denote by ϕj,k,h1,h2 the corresponding eigenvectors.

Let us now introduce the following classes of solutions of (5.114) for any 0 < γ < 1:

Cγ(h) = span ϕj,k,h1,h2 such that |λj,k,h1,h2|max(h1, h2) ≤ 2√γ.

The following Lemma holds (see [108]):

Lemma 5.5.2. Let 0 < γ < 1. Then there exist Tγ such that for all T > Tγ there exist

kT,γ > 0 and KT,γ > 0 such that

kT,γEh(0) ≤∫ T

0‖BhU(t)‖2

Yhdt ≤ KT,γEh(0) (5.119)

holds for every solution of (5.114) in the class Cγ(h) and every h1, h2 small enough satisfying

sup∣∣∣h1

h2

∣∣∣ <√

γ

4 − γ.

Now we present the time discrete schemes we are interested in. For any t > 0, we

consider the following time Newmark approximation scheme of system (5.114):

Uk+1 + Uk−1 − 2Uk

(t)2 +Ah

(βUk+1 + (1 − 2β)Uk + βUk−1

)= 0,

(U0 + U1

2,U1 − U0

t)

= (Uh,0, Uh,1),

(5.120)


with β ≥ 1/4.

The energy of (5.120) given by

Ek =1

2

∥∥∥∥A12h

(Uk + Uk+1

2

)∥∥∥∥2

+1

2

∥∥∥∥Uk+1 − Uk

t

∥∥∥∥2

+ (4β − 1)(t)2

8

∥∥∥∥A12h

(Uk+1 − Uk

t)∥∥∥∥

2

, k ∈ Z (5.121)

which is a discrete counterpart of the time continuous energy (5.54) and remains constant

(see (5.62) as well).

In view of (5.119), conditions (B1) and (B2) are satisfied. Besides, conditions (A1)

and (A2) are straightforward. Therefore the following theorem can be obtained as a direct

consequence of Theorem 5.3.2:

Theorem 5.5.1. Set 0 < γ < 1. Assume that the mesh sizes h1, h2 and t tend to zero

and

sup∣∣∣h1

h2

∣∣∣ <√

γ

4 − γ,

maxh1, h2t ≤ τ, (5.122)

where τ is a positive constant.

Then, for any 0 < δ ≤ 2√γ/τ , there exist Tδ > 0 such that for any T > Tδ, there exists

kT,δ,γ > 0 such that the observability inequality

kT,δ,γEk ≤ t

∑

kt∈(0,T )

∥∥∥BhUk∥∥∥

2

Yh

holds for every solution of (5.120) with initial data in the class Cδ/t for h1, h2,t small

enough satisfying (5.122).

Proof. We are in the setting given before and thus Lemma 5.5.1 applies. Hence, to apply

Theorem 5.3.1, we only need to verify that Cδ/t ⊂ Cγ(h). But

|λ| < δ

t ⇒ |λ| ≤ 2

√γ

τt ≤ 2

√γ

maxh1, h2.

and this completes the proof. ♦

The 1-d string with rapidly oscillating density

In this paragraph, we consider a one-dimensional wave equation with rapidly oscillating

density, which provides another example where the model under consideration depends on

an extra parameter.


Let us state the problem. Let ρ ∈ L∞(lR) be a periodic function such that 0 < ρm ≤ρ(x) ≤ ρM < ∞, a.e. x ∈ lR. Given ε > 0, set ρε(x) = ρ(x/ε) and consider the one-

dimensional wave equationρε(x)λε

nΦεn + ∂2

xxΦεn = 0, x ∈ (0, 1),

Φεn(0) = Φε

n(1) = 0.(5.123)

We consider the observation operator

Bεuε(t) = ∂xuε(1, t). (5.124)

The mathematical setting is the same as in Subsection 5.4.3 and therefore we do not recall

it.

The eigenvalue problem for (5.123) reads

ρε(x)λεnΦε

n + ∂2xxΦε

n = 0, x ∈ (0, 1); Φεn(0) = Φε

n(1) = 0. (5.125)

For each ε > 0, there exists a sequence of eigenvalues

0 < λε1 < λε

2 < · · · < λεn < · · · → ∞

and a sequence of associated eigenfunctions (Φεn)n which can be chosen to constitute an

orthonormal basis in L2(0, 1) with respect to the norm

‖φ‖2L2 =

∫ 1

0ρε(x)|φ(x)|2 dx.

In [4], the following is proved:

Theorem 5.5.2 ([4]). There exists a positive number D > 0, such that the following holds:

Let T > 2√ρ, where ρ denotes the mean value of ρ. Then there exist two positive

constants kT and KT such that for any initial data (u0, v0) in

CD/ε = span Φεn : n < D/ε,

the solution uε of (5.123) verifies

kT ‖(u0, v0)‖2H1

0 (0,1)×L2(0,1) ≤∫ T

0|uε

x(1, t)|2dt ≤ KT ‖(u0, v0)‖2H1

0 (0,1)×L2(0,1) .

Let us then consider the following time semi-discretization of (5.123)

ρε(x)(uε,k+1 − 2uε,k + uε,k−1

(t)2)− ∂2

xx

((1 − 2β)uε,k + β(uε,k−1 + uε,k+1)

)= 0,

(x, k) ∈ (0, 1) × lN, (5.126)


completed with the following boundary conditions and initial data

uε,k(0) = uε,k(1) = 0, k ∈ lN,uε,0 + uε,1

2(x) = u0(x),

uε,1 − uε,0

t (x) = v0(x), x ∈ (0, 1).(5.127)

Since conditions (A1)-(A2)-(B1)-(B2) hold, we get the following result as a consequence

of Theorem 5.3.2:

Theorem 5.5.3. Let δ > 0 and β > 1/4. Assume that the parameters t and ε tend to

zero and

δε ≤ Dt.

Then there exists a time Tδ such that for any T > Tδ, there exists a positive constant

kT,δ such that the observability inequality

kT,δ ‖(u0, v0)‖2H1

0 (0,1)×L2(0,1) ≤ t∑

kt∈(0,T )

|uε,kx (1)|2 (5.128)

holds for every solution of (5.126)-(5.127) with initial data (u0, v0) in the class Cδ/t, in-

dependently of t and ε.

5.6. On the admissibility condition

The goal of this section is to provide admissibility results for the time-discrete schemes

used throughout this Chapter. These results are complementary to the observability results

proved in Theorems 5.2.1, 5.3.1 and 5.3.2 when dealing with controllability problems (see

[56]).

5.6.1. The time-continuous setting

Let us assume that system (2.18)-(2.19) is admissible. By definition, there exists a

positive constant KT such that :

∫ T

0‖y(t)‖2

Y dt ≤ KT ‖z0‖2X ∀ z0 ∈ D(A). (5.129)

The goal of this section is to prove that this property can be read on the wave packets

setting as well.

5.6. ON THE ADMISSIBILITY CONDITION 137

Proposition 5.6.1. System (2.18)-(2.19) is admissible if and only if

There exists r > 0 and D > 0 such that

for all n ∈ Λ and for all z =∑

l∈Jr(µn)

clΦl : ‖Bz‖Y ≤ D ‖z‖X , (5.130)

where

Jr(µ) = l ∈ lN, such that |µl − µ| ≤ r. (5.131)

Proof. We will prove separately the two implications.

First let us assume that system (2.18)-(2.19) is admissible.

Denote by

V (ω, ε) = span(Φj such that |µj − ω| ≤ ε

).

Then the following lemma holds:

Lemma 5.6.1. Let us define K(ω, ε) as

K(ω, ε) =∥∥B(A− iωI)−1

∥∥L(V (ω,ε)∗,Y )

.

Then for any ε > 0, K(ω, ε) is uniformly bounded in ω, that is

K(ε) = supω∈lR

K(ω, ε) <∞. (5.132)

Besides, the following estimate holds

K(ε) ≤√

K1

1 − exp(−1)

(1 +

1

ε

), (5.133)

where K1 is the admissibility constant in (2.20).

Proof of Lemma 5.6.1.

Let us first notice these resolvent identities:

(A− iωI) − I = A− (1 + iω)I

(A− (1 + iω)I)−1(I − (A− iωI)−1) = (A− iωI)−1.

Hence

K(ω, ε) ≤∥∥B(A− (1 + iω)I)−1

∥∥L(X,Y )

∥∥(I − (A− iωI)−1)∥∥

L(V (ω,ε)∗,X).


Obviously∥∥(I − (A− iωI)−1)

∥∥L(V (ω,ε)∗,X)

≤ 1 +1

ε

Hence we restrict ourselves to the study of

∥∥B(A− (1 + iω)I)−1∥∥

L(X,Y ).

Let us remark that for all z =∑ajΦj ∈ X,

(A− (1 + iω)I)−1z =∑ 1

i(µj − ω) − 1ajΦj

=

∫ ∞

0exp(−(1 + iω)t)z(t) dt,

where z(t) is the solution of (2.18) with initial value z. This implies that

∥∥B(A− (1 + iω)I)−1z∥∥2

Y=

∥∥∥∥∫ ∞

0exp(−1 − iωt)Bz(t) dt

∥∥∥∥2

Y

≤∫ ∞

0

∣∣∣ exp(−1 − 2iωt)∣∣∣ dt

(∫ ∞

0exp(−t) ‖Bz(t)‖2

Y dt)

≤∫ ∞


Y dt.

But using the admissibility property of the operator B, we obtain∫ ∞


Y dt ≤∑

k∈lN

exp(−k)∫ k+1

k‖Bz(t)‖2

Y dt

≤(∑

k∈lN

exp(−k))K1 ‖z‖2

X

≤ K1

1 − exp(−1)‖z‖2

X .

The estimate (5.133) follows.♦

Let us now consider a wave packet z0 =∑

l∈J1(µn) clΦl. Then taking ε = 1 in Lemma

5.6.1, one gets that

‖Bz‖Y ≤∥∥B(A− i(µn − 2)I)−1

∥∥L(V (µn−2,1)∗,Y )

‖(A− i(µn − 2)I)z‖

≤ K(1)(

maxl∈J1(µn)

|µl − µn| + 1)‖z‖

≤ 3K(1) ‖z‖ .

Now we assume that estimate (5.130) holds for some r > 0 and D > 0. Set z0 ∈ D(A),

and expand z0 as

z0 =∑

k∈Z

zk, zk =∑

l∈Jr(2kr)

clΦl.


We need a special test function whose existence is established in the following Lemma:

Lemma 5.6.2. There exists a time T and a function M satisfying

M(t) ≥ 0, |t| ≥ T/2,

M(t) ≥ 1, |t| ≤ T/2,

Supp M ⊆ (−2r, 2r).

(5.134)

The proof is postponed to the end of this section. Note that functions satisfying similar

properties appear naturally in the proofs of various Ingham’s type inequalities, see [33] and

[95].

Taking Lemma 5.6.2 into account, we estimate

∫ T

0‖Bz(t)‖2

Y ≤∫

lRM(t− T/2) ‖Bz(t)‖2

Y dt

≤∑

k1,k2

∫

lRM(t− T/2) < Bzk1(t), Bzk2(t) >Y ×Y dt.

But these scalar products vanish most of the time. Indeed, if |k1 − k2| ≥ 2, from (5.134),

we get

∫

lRM(t− T/2) < Bzk1(t), Bzk2(t) >Y dt

=∑

(l1,l2)∈Jr(2k1r)×Jr(2k2r)

M(µl1 − µl2) < al1BΦl1, al2BΦl2 >Y = 0.

This implies that

∫ T

0‖Bz(t)‖2

Y≤∫

lRM(t− T/2)

∑

k

(‖Bzk(t)‖2

Y + 2Re < Bzk(t), Bzk+1(t) >Y ×Y

)dt

≤ 3

∫

lRM(t− T/2)

∑

k

‖Bzk(t)‖2Y dt

≤ 3D

∫

lRM(t− T/2)

∑

k

‖zk(t)‖2X dt

≤ 3DM (0) ‖z0‖2X .

This completes the proof, since admissibility at time T is obviously equivalent to admissi-

bility in any time T ≥ T .♦

Proof of Lemma 5.6.2.

In this proof, we do not care about the value of the parameters r and T that can be

handled through a rescaling argument.


Let us consider the function

f(t) =1

πsinc (t) =

sin(t)

πt.

It is well-known that its Fourier transform is f(τ) = χ(−1,1)(τ), where χ(−1,1) denotes the

characteristic function of (−1, 1).

Hence, the function

M(t) = f(t)2 =sinc 2(t)

π2

satisfies the following properties

M(t) ≥ 2

π3, |t| < π

4; M(t) ≥ 0, t ∈ lR; M (τ) = (2 − |τ |)+, τ ∈ lR

and the proof is complete. For instance, for r > 0, one can take the function Mr(t) as

Mr(t) =π2

8sinc 2(rt) (5.135)

which satisfies (5.134) with T = π/2r. ♦

Remark 5.6.1. In the context of families of pairs (A,B), according to Proposition 5.6.1,

the uniform admissibility condition (5.107) is equivalent to a uniform wave packet estimate

similar to (5.130). To be more precise, if (Φpj)j∈lN denotes the eigenvectors of Ap associated

to the eigenvalues (λpj )j∈lN, that is ApΦ

pj = λp

jΦpj , the uniform admissibility condition is

equivalent to:

There exists r > 0 and D > 0 such that for all p, n ∈ lN

and for all z =∑

l∈Jr(λpn)

clΦpl : ‖Bpz‖Yp

≤ D ‖z‖Xp.

5.6.2. The time-discrete setting

This subsection is aimed to prove that if the continuous system (2.18)-(2.19) is admis-

sible, in the sense of Definition 2.5.1, then its time semi-discrete approximations systems

will be admissible as well under suitable assumptions. In this part, we will focus on the

particular discretization given in Subsection 5.3.1, but everything works as well in all the

time semi-discretization schemes considered in this Chapter.

More precisely, we assume that the continuous system (2.18)-(2.19) is admissible, that

is, from Proposition 5.6.1, the wave packet estimate (5.130) holds.

Then we claim that, under the assumptions (5.34), (5.35), (5.36), (5.37) and (5.38), the

following discrete admissibility inequality holds:


Theorem 5.6.1. Assume that system (2.18)-(2.19) is admissible. Set δ > 0. For any

T > 0, there exists a constant KT,δ > 0 such that for all t small enough, the solution of

equation (5.34) with initial data in Cδ/t satisfies

tT/t∑

k=0

∥∥∥Bzk∥∥∥

2

Y≤ KT,δ

∥∥z0∥∥2

X. (5.136)

Proof. The proof follows the one given in the continuous case. First of all, let us remark

the following straightforward fact: There exists rδ > 0 such that for all n ∈ Z satisfying

t|λn,t| ≤ δ, for all t > 0, the set

Jrδ(λn,t) = l ∈ Z, such that |λl,t − λn,t| ≤ rd,

where λl,t is as in (5.36), is a subset of Jr(µn) (recall (5.131)). Besides, we have the

following estimate on rδ :

rδ ≤ r inf|h′(η)|, |η| ≤ δ.

Note that condition (5.37) implies the positivity of the right hand side.

Given t > 0, assume that there is a time T and a function Mt ∈ l2(tZ) such that

Mt,k ≥ 0, |kt| ≥ T/2,

Mt,k ≥ 1, |t| ≤ T/2,

Supp Mt ⊆ (−2rδ, 2rδ),

(5.137)

where this time Mt denotes the discrete Fourier transform at scale t defined in Definition

5.2.1. One can easily check that we can take Mt = Mrδfor all t > 0 where Mrδ

is as in

(5.135).

With this definition, the proof of inequality (5.136) consists in rewriting the one of

Proposition 5.6.1 by replacing the continuous integrals and the Fourier transform by their

discrete versions. Since all the steps are independent of t, the admissibility inequality

holds uniformly. ♦

Note that this proof can be applied to derive uniform admissibility results for families of

operators (A,B) within the class D(CB , T, kT ,KT ) for the fully discrete schemes. Indeed,

in the setting of Section 5.5, according to Remark 5.6.1, the proof presented above directly

implies uniform admissibility properties for operators in the class D(CB , T, kT ,KT ) when

the initial data are taken in the filtered class Cδ/t.

Chapter 6

Conclusions and open problems

In this Thesis we have analyzed the following problems:

1. The interior control and observation of the time-discrete heat equation.

2. The boundary controllability and observability of the time-discrete wave equation.

3. The observability of time-discrete approximations of abstract conservative systems.

For the heat equation, we have studied the null-controllability and observability of an

implicit Euler time discretization in a bounded domain with a local internal controller.

Using Lebeau-Robbiano’s time iteration method, we have proved that the projection in an

appropriate filtered space is null controllable with uniformly bounded controls with respect

to the time-step parameter. In this way, the well-known null-controllability property of the

time-continuous heat equation can be proven as the limit, as t→ 0, of the controllability

of projections of the time-discrete one.

Analogous results have been also obtained for other discrete schemes, as the explicit

Euler scheme, the θ-method, etc. We have also discussed the null-controllability of the

implicit Euler time-discrete parabolic equation of fractional order, generalizing previous

results in [67] in the continuous setting.

For the wave equation we have studied the exact boundary controllability of a trapezoidal

time discretization in a bounded domain. We have proved, using multiplier techniques, that

the projection of the solution in an appropriate filtered space is exactly controllable with

uniformly bounded cost with respect to the time-step. In this way, the well-known exact-

controllability property of the time-continuous wave equation has been reproduced as the

limit, as the time step t→ 0, of the controllability of projections of the time-discrete one.

143

144 CHAPTER 6. CONCLUSIONS AND OPEN PROBLEMS

By duality these results are equivalent to deriving uniform observability estimates (with

respect to t → 0) within a class of solutions of the time-discrete problem in which the

high frequency components have been filtered. The later has been established by means

of a time-discrete version of the classical multiplier technique. The optimality of the order

of the filtering parameter has also been established, although a careful analysis of the

expected velocity of propagation of time-discrete waves indicates that its exact value could

be improved.

The last result of this Thesis concerns the uniform observability of various time dis-

cretizations of an abstract evolution system zt = Az, where A is a skew-adjoint operator,

and an observation operator B is given. More precisely, we assume that the pair (A,B) is

exactly observable in the continuous setting, and we derive uniform observability inequal-

ities for some discretization schemes provided we filter the initial data. The method we

use is mainly based on the resolvent estimate given in [3], which turns out to be equivalent

to the exact observability property of the pair (A,B). We present some applications of

our results to time-discrete and fully-discrete approximation schemes. In particular, the

uniform boundary observability of the time-discrete wave equation, which has been shown

in Chapter 4, can be seen as a direct consequence of the abstract results in this chapter.

In the following we present some comments and open problems related to the subjects

and problems treated in this Thesis.

Optimal order of the filtering parameter of the time-discrete heat equation.

In Chapter 3 we prove that the projection of the solution of the time-discrete heat

equation (3.1) is uniformly controllable, within the subspace generated by the eigen-

functions corresponding to eigenvalues λ ≤ (t)−r with 0 < r < 2. The restriction

r < 2 has been imposed for technical reasons. However, it is likely that, when r = 2,

the result fails because of the lack of sufficient dissipation, as it happens in the critical

fractional order heat equation (see [67]). This is an interesting open problem.

Uniform boundary controllability of the time-discrete heat equation.

The results of Chapter 3 concern the interior control problem. The same issues make

sense in the context of boundary controllability. Recall that the time-continuous heat

145

equation is controllable for all time T and an arbitrarily support Γ0, open nonempty

subset of Γ (see, for instance, [50]). However, even in the time continuous case, one

can not derive the boundary controllability directly by means of the L-R method since

(2.10) is false when the observation set is a subset of the boundary. This is obvious,

in particular, in the 1 − d case.

A rather standard method to derive boundary controllability out of interior control-

lability is based on extension-restriction arguments and it is as follows. One first

extends by zero the solution to an outer neighborhood of Γ0. The arguments for in-

terior control allow to control the system in the whole domain by means of a control

supported in this small added domain. The restriction of the solution to the original

domain satisfies all of the requirements and its restriction or trace to the subset of

the boundary where the control had to be supported, provides the boundary control

we are looking for.

This technique can be combined with the filtering arguments we have developed for

the problem of interior control. But the results obtained this way are hard to interpret

because the eigenfunctions of the extended domain enter in it. Very likely, in this con-

text, the most promising technique is that based on the use of Carleman inequalities.

But so far these inequalities have not been developed in the time-discrete setting.

Full-discrete approximations of the heat equation and their control prob-

lems.

In Chapter 3, we have analyzed time semi-discrete schemes for the heat equation and

their control properties. Naturally, as a further step, one could consider the control

problem for fully discrete schemes, both on time and space variables.

For instance, let Ω = (0, 1) and ω = (a, b) with 0 < a < b < 1. For any given J ∈ lN∗we set h = 1/(J + 1) and introduce the net x0 = 0 < x1 < · · · < xN+1 = 1 with

xj = jh and j = 0, 1, · · · , J + 1.

A full implicit discretization scheme for system (1.1) reads:

yk+1j − yk

j

t −yk+1

j+1 + yk+1j−1 − 2yk+1

j

(x)2 = ukj 1[a,b], j = 0, 1, · · · , J + 1,

k = 0, 1, · · · ,K − 1

yk0 = yk

J+1 = 0, k = 1, · · · ,K(y0

0, · · · , y0J+1) given.

(6.1)


In (6.1) (ykj ) indicates the approximation of y(kt, jh) and uk

j 1[a,b] formally indicates

the approximation of u(kt, jh)1ω .

As far as the authors know, the controllability of this fully discrete scheme is an

open problem. At this level it is very likely that one possible method to be explored

could be the use of time-discrete biorthogonal sequences, as a discrete counterpart

of the existing theory for time-continuous 1 − d parabolic problems ([17]). Another

possibility could be the transmutation method ([68]) via which one can transfer the

uniform controllability results on space semi-discrete wave equations ([64], [110]) to

the fully discrete scheme (6.1).

Optimality of the filtering parameter in the uniform boundary observabil-

ity of the time-discrete wave equation.

In Chapter 4, for the uniform observability to hold, we have chosen the filtering

parameter of the form δ(t)−2, with

0 < δ < min( 4√

d2 + 16a2 + d,

2a1T + 4R − 2T

a3 +√a2

3 − 4Ta2(a1T + 2R− T )

),

where a1, a2 and a3 are constants depending on d, T and R.

However, when applying Theorem 5.2.1, δ can be chosen to be arbitrarily large, pro-

vided Tδ is taken large enough. This fact is in agreement with the analysis we have

done of the velocity of propagation along bicharacteristic rays.

It is an open problem to identify the optimal observation/control time Tδ, as a function

of δ.

Optimal observability time of abstract time-discrete conservative systems.

To our knowledge, the time of observability that the resolvent estimate in Theorem

2.5.1 yields is not optimal in general. The same can be said about the wave-packet

method and the viewpoint adopted in Theorem 2.5.2.

In the continuous context, the proofs of those results need to be improved to derive

observability inequalities in optimal times. The same is also to be done at the time-

discrete level.

Time-discrete Carleman estimate.

One of the most commonly used technique to obtain observability inequalities for wave

and heat equations and some other models are the so-called Carleman inequalities

147

([21], [23], [29], [30], [31], [91] and the references therein). As far as we know they

have not been adapted to time-discrete or fully discrete equations.

Bibliography

[1] Bardos, C., Lebeau, G. and Rauch, J., Un exemple d’utilisation des notions de prop-

agation pour le controle et la stabilisation de problemes hyperboliques, Rend. Sem.

Mat. Univ. Politec. Torino, (Special Issue), 11–31 (1988).

[2] ——, Sharp sufficient conditions for the observation, control, and stabilization of

waves from the boundary, SIAM J. Control Optim., 30 (5), 1024–1065 (1992).

[3] Burq, N. and Zworski, M., Geometric control in the presence of a black box, J. Amer.

Math. Soc., 17 (2), 443–471 (electronic) (2004).

[4] Castro, C., Boundary controllability of the one-dimensional wave equation with

rapidly oscillating density, Asymptot. Anal., 20 (3-4), 317–350 (1999).

[5] Castro, C. and Micu, S., Boundary controllability of a linear semi-discrete 1-D wave

equation derived from a mixed finite element method, Numer. Math., 102 (3), 413–462

(2006).

[6] Cazenave, T. and Haraux, A., An introduction to semilinear evolution equations,

vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon

Press Oxford University Press, New York, 1998.

[7] Coron, J.-M. and Crepeau, E., Exact boundary controllability of a nonlinear KdV

equation with critical lengths, J. Eur. Math. Soc. (JEMS), 6 (3), 367–398 (2004).

[8] Dager, R., Observation and control of vibrations in tree-shaped networks of strings,

SIAM J. Control Optim., 43 (2), 590–623 (electronic) (2004).

[9] ——, Insensitizing controls for the 1-D wave equation, SIAM J. Control Optim.,

45 (5), 1758–1768 (electronic) (2006).

149

150 BIBLIOGRAPHY

[10] Dager, R. and Zuazua, E., Wave propagation, observation and control in 1-d flexible

multi-structures, vol. 50 of Mathematiques & Applications (Berlin) [Mathematics &

Applications], Springer-Verlag, Berlin, 2006.

[11] Delfour, M. C. and Mitter, S. K., Controllability and observability for infinite-

dimensional systems, SIAM J. Control, 10, 329–333 (1972).

[12] Ervedoza, S., On the mixed finite-element method for the 1-d wave equation on non-

uniform meshes, Preprint (2007).

[13] Ervedoza, S. and Zuazua, E., Perfectly matched layers in 1-d: Energy decay for

continuous and semi-discrete waves, Preprint (2007).

[14] Evans, L. C., Partial differential equations, vol. 19 of Graduate Studies in Mathemat-

ics, American Mathematical Society, Providence, RI, 1998.

[15] Fattorini, H. O., Boundary control systems, SIAM J. Control, 6, 349–385 (1968).

[16] Fattorini, H. O. and Russell, D. L., Exact controllability theorems for linear parabolic

equations in one space dimension, Arch. Rational Mech. Anal., 43, 272–292 (1971).

[17] ——, Uniform bounds on biorthogonal functions for real exponentials with an appli-

cation to the control theory of parabolic equations, Quart. Appl. Math., 32, 45–69

(1974/75).

[18] Fernandez-Cara, E. and Zuazua, E., The cost of approximate controllability for heat

equations: The linear case, Adv. Diff. Eqs., 5, 465–514 (2000).

[19] ——, Control theory: History, mathematical achievements and perspectives, Boletı

de SEMA, (26) (2003).

[20] ——, On the history and perspectives of control theory, Matapli, (74), 47–73 (2004).

[21] Fu, X., A weighted identity for partial differential operators of second order and its

applications, C. R. Math. Acad. Sci. Paris, 342 (8), 579–584 (2006).

[22] Fu, X., Zhang, X. and Zuazua, E., On the optimality of some observability inequal-

ities for plate systems with potentials, in Phase space analysis of partial differen-

tial equations, vol. 69 of Progr. Nonlinear Differential Equations Appl., pp. 117–132,

Birkhauser Boston, Boston, MA, 2006.

BIBLIOGRAPHY 151

[23] Fursikov, A. V. and Imanuvilov, O. Y., Controllability of evolution equations, vol. 34

of Lecture Notes Series, Seoul National University Research Institute of Mathematics

Global Analysis Research Center, Seoul, 1996.

[24] Glowinski, R., Ensuring well-posedness by analogy: Stokes problem and boundary

control for the wave equation, J. Comput. Phys., 103 (2), 189–221 (1992).

[25] Glowinski, R., Li, C. H. and Lions, J.-L., A numerical approach to the exact boundary

controllability of the wave equation. I. Dirichlet controls: description of the numerical

methods, Japan J. Appl. Math., 7 (1), 1–76 (1990).

[26] Guillemin, V., Some classical theorems in spectral theory revisited, in Seminar on

Singularities of Solutions of Linear Partial Differential Equations (Inst. Adv. Study,

Princeton, N.J., 1977/78), vol. 91 of Ann. of Math. Stud., pp. 219–259, Princeton

Univ. Press, Princeton, N.J., 1979.

[27] Hairer, E., Nørsett, S. P. and Wanner, G., Solving ordinary differential equations.

I, vol. 8 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin,

1993, 2nd ed.

[28] Hairer, E. and Wanner, G., Solving ordinary differential equations. II, vol. 14 of

Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1991.

[29] Imanuvilov, O. Y., On Carleman estimates for hyperbolic equations, Asymptot. Anal.,

32 (3-4), 185–220 (2002).

[30] Imanuvilov, O. Y. and Puel, J.-P., Global Carleman estimates for weak solutions

of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., (16), 883–913

(2003).

[31] Imanuvilov, O. Y. and Yamamoto, M., Carleman inequalities for parabolic equations

in Sobolev spaces of negative order and exact controllability for semilinear parabolic

equations, Publ. Res. Inst. Math. Sci., 39 (2), 227–274 (2003).

[32] Infante, J.-A. and Zuazua, E., Boundary observability for the space-discretizations of

the 1-d wave equation, C. R. Acad. Sci. Paris Ser. I Math., 326 (6), 713–718 (1998).

[33] Ingham, A. E., Some trigonometrical inequalities with applications to the theory of

series, Math. Z., 41 (1), 367–379 (1936).

152 BIBLIOGRAPHY

[34] Isaacson, E. and Keller, H. B., Analysis of numerical methods, John Wiley & Sons

Inc., New York, 1966.

[35] Iserles, A., A first course in the numerical analysis of differential equations, Cambridge

Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1996.

[36] Iserles, A. and Nørsett, S. P., Biorthogonal polynomials in numerical ODEs, Ann.

Numer. Math., 1 (1-4), 153–170 (1994).

[37] Iyengar, K. S. K., Madhava Rao, B. S. and Nanjundiah, T. S., Some trigonometrical

inequalities, Half-Yearly J. Mysore Univ. Sect. B., N. S., 6, 1–12 (1945).

[38] Jaffard, S., Tucsnak, M. and Zuazua, E., On a theorem of Ingham, J. Fourier Anal.

Appl., 3 (5), 577–582 (1997), dedicated to the memory of Richard J. Duffin.

[39] Jerison, D. and Lebeau, G., Nodal sets of sums of eigenfunctions, in Harmonic analysis

and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Math., pp.

223–239, Univ. Chicago Press, Chicago, IL, 1999.

[40] Kalman, R. E., Contributions to the theory of optimal control, Bol. Soc. Mat. Mexi-

cana (2), 5, 102–119 (1960).

[41] ——, On the general theory of control systems, Proceedings of the First IFAC

Congress on Automatic Control, 1, 481–492 (1961).

[42] Kalman, R. E., Ho, Y. C. and Narendra, K. S., Controllability of linear dynamical

systems, Contributions to Differential Equations, 1, 189–213 (1963).

[43] Kannai, Y., Off diagonal short time asymptotics for fundamental solutions of diffusion

equations, Commun. Partial Differ. Equations, 2 (8), 781–830 (1977).

[44] Knobel, R., An introduction to the mathematical theory of waves, vol. 3 of Stu-

dent Mathematical Library, American Mathematical Society, Providence, RI, 2000,

iAS/Park City Mathematical Subseries.

[45] Krabs, W., On moment theory and controllability of one-dimensional vibrating sys-

tems and heating processes, vol. 173 of Lecture Notes in Control and Information

Sciences, Springer-Verlag, Berlin, 1992.

BIBLIOGRAPHY 153

[46] Kravanja, P. and Van Barel, M., Computing the zeros of analytic functions, vol. 1727

of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000.

[47] Lebeau, G., Controle analytique. I. Estimations a priori, Duke Math. J., 68 (1), 1–30

(1992).

[48] ——, Controle de l’equation de Schrodinger, J. Math. Pures Appl. (9), 71 (3), 267–291

(1992).

[49] Lebeau, G. and Robbiano, L., Controle exact de l’equation de la chaleur, Comm.

Partial Differential Equations, 20 (1-2), 335–356 (1995).

[50] ——, Controle exacte de l’equation de la chaleur, in Seminaire sur les Equations aux

Derivees Partielles, 1994–1995, pp. Exp. No. VII, 13, Ecole Polytech., Palaiseau, 1995.

[51] ——, Stabilization of the wave equation by the boundary, in Partial differential equa-

tions and mathematical physics (Copenhagen, 1995; Lund, 1995), vol. 21 of Progr.

Nonlinear Differential Equations Appl., pp. 207–210, Birkhauser Boston, Boston, MA,

1996.

[52] Lebeau, G. and Zuazua, E., Null-controllability of a system of linear thermoelasticity,

Arch. Rational Mech. Anal., 141 (4), 297–329 (1998).

[53] Lee, E. B. and Markus, L., Foundations of optimal control theory, Robert E. Krieger

Publishing Co. Inc., Melbourne, FL, 1986, 2nd ed.

[54] Li, X., Sun, L. and Cheng, Y., Control Theory in the Computer Application, Lecture

Notes Series, Fundan University Press, 1987.

[55] Lions, J.-L., Controlabilite exacte, perturbations et stabilisation de systemes dis-

tribues. Tome 1, vol. 8 of Recherches en Mathematiques Appliquees [Research in

Applied Mathematics], Masson, Paris, 1988.

[56] ——, Controlabilite exacte, perturbations et stabilisation de systemes distribues.

Tome 2, vol. 9 of Recherches en Mathematiques Appliquees [Research in Applied

Mathematics], Masson, Paris, 1988.

[57] ——, Exact controllability, stabilization and perturbations for distributed systems,

SIAM Rev., 30 (1), 1–68 (1988).

154 BIBLIOGRAPHY

[58] ——, Remarks on approximate controllability, J. Anal. Math., 59, 103–116 (1992).

[59] Lions, J.-L. and Magenes, E., Problemes aux limites non homogenes et applications.

Vol. 1, Travaux et Recherches Mathematiques, No. 17, Dunod, Paris, 1968.

[60] Lopez, A. and Zuazua, E., Some new results related to the null-controllability of the

1-d heat equation, in Seminaire sur les Equations aux Derivees Partielles, 1997–1998,

pp. Exp. No. VIII, 22, Ecole Polytech., Palaiseau, 1998.

[61] Macia, F., The effect of group velocity in the numerical analysis of control problems

for the wave equation, in Mathematical and numerical aspects of wave propagation—

WAVES 2003, pp. 195–200, Springer, Berlin, 2003.

[62] Macia, F. and Zuazua, E., On the lack of observability for wave equations: a Gaussian

beam approach, Asymptot. Anal., 32 (1), 1–26 (2002).

[63] Markus, L., Controllability of nonlinear processes, J. Soc. Indust. Appl. Math. Ser. A

Control, 3, 78–90 (1965).

[64] Micu, S., Uniform boundary controllability of a semi-discrete 1-D wave equation,

Numer. Math., 91 (4), 723–768 (2002).

[65] Micu, S. and Tucsnak, M., Approximate controllability of a semi-discrete 1-D wave

equation, An. Univ. Craiova Ser. Mat. Inform., 32, 48–58 (2005).

[66] Micu, S. and Zuazua, E., An introduction to the controllability of partial differential

equations, in “Quelques questions de theorie du controle”, Sari, T., ed., pp. 69–157,

Collection Travaux en Cours Hermann, 2004.

[67] ——, On the controllability of a fractional order parabolic equation, SIAM J. Control

Optim., 44 (6), 1950–1972 (electronic) (2006).

[68] Miller, L., Geometric bounds on the growth rate of null-controllability cost for the

heat equation in small time, J. Differential Equations, 204 (1), 202–226 (2004).

[69] ——, How violent are fast controls for Schrodinger and plate vibrations?, Arch. Ra-

tion. Mech. Anal., 172 (3), 429–456 (2004).

[70] ——, Controllability cost of conservative systems: resolvent condition and transmu-

tation, J. Funct. Anal., 218 (2), 425–444 (2005).

BIBLIOGRAPHY 155

[71] ——, Unique continuation estimates for the Laplacian and the heat equation on non-

compact manifolds, Math. Res. Lett., 12 (1), 37–47 (2005).

[72] ——, On the cost of fast controls for thermoelastic plates, Asymptot. Anal., 51 (2),

93–100 (2007).

[73] Munch, A., A uniformly controllable and implicit scheme for the 1-D wave equation,

M2AN Math. Model. Numer. Anal., 39 (2), 377–418 (2005).

[74] Negreanu, M. and Zuazua, E., Uniform boundary controllability of a discrete 1-D

wave equation, Systems Control Lett., 48 (3-4), 261–279 (2003), optimization and

control of distributed systems.

[75] ——, Convergence of a multigrid method for the controllability of a 1-d wave equation,

C. R. Math. Acad. Sci. Paris, 338 (5), 413–418 (2004).

[76] ——, Discrete Ingham inequalities and applications, C. R. Math. Acad. Sci. Paris,

338 (4), 281–286 (2004).

[77] ——, Discrete Ingham inequalities and applications, SIAM J. Numer. Anal., 44 (1),

412–448 (electronic) (2006).

[78] Orive, R., Weakly nonlinear long-time behavior of solutions to a hyperbolic relaxation

systems, in EQUADIFF 2003, pp. 666–671, World Sci. Publ., Hackensack, NJ, 2005.

[79] Orive, R. and Zuazua, E., Finite difference approximation of homogenization problems

for elliptic equations, Multiscale Model. Simul., 4 (1), 36–87 (electronic) (2005).

[80] ——, Long-time behavior of solutions to a nonlinear hyperbolic relaxation system, J.

Differential Equations, 228 (1), 17–38 (2006).

[81] Orive, R., Zuazua, E. and Pazoto, A. F., Asymptotic expansion for damped wave

equations with periodic coefficients, Math. Models Methods Appl. Sci., 11 (7), 1285–

1310 (2001).

[82] Ortega, R. and Spong, M. W., Adaptive motion control of rigid robots: a tutorial,

Automatica J. IFAC, 25 (6), 877–888 (1989).

[83] Osses, A., A rotated multiplier applied to the controllability of waves, elasticity,

and tangential Stokes control, SIAM J. Control Optim., 40 (3), 777–800 (electronic)

(2001).

156 BIBLIOGRAPHY

[84] Ramdani, K., Takahashi, T., Tenenbaum, G. and Tucsnak, M., A spectral approach

for the exact observability of infinite-dimensional systems with skew-adjoint generator,

J. Funct. Anal., 226 (1), 193–229 (2005).

[85] Rosier, L., Exact boundary controllability for the Korteweg-de Vries equation on a

bounded domain, ESAIM Control Optim. Calc. Var., 2, 33–55 (electronic) (1997).

[86] Russell, D. L., A unified boundary controllability theory for hyperbolic and parabolic

partial differential equations, Studies in Appl. Math., 52, 189–211 (1973).

[87] ——, Controllability and stabilizability theory for linear partial differential equations:

recent progress and open questions, SIAM Rev., 20 (4), 639–739 (1978).

[88] Russell, D. L. and Zhang, B. Y., Controllability and stabilizability of the third-order

linear dispersion equation on a periodic domain, SIAM J. Control Optim., 31 (3),

659–676 (1993).

[89] Safarov, Y. and Vassiliev, D., The asymptotic distribution of eigenvalues of partial

differential operators, vol. 155 of Translations of Mathematical Monographs, American

Mathematical Society, Providence, RI, 1997.

[90] Sussmann, H. J., A general theorem on local controllability, SIAM J. Control Optim.,

25 (1), 158–194 (1987).

[91] Tang, S. and Zhang, X., Carleman inequality for backward stochastic parabolic equa-

tions with general coefficients, C. R. Math. Acad. Sci. Paris, 339 (11), 775–780 (2004).

[92] Taylor, M. E., Partial differential equations. I, vol. 115 of Applied Mathematical

Sciences, Springer-Verlag, New York, 1996.

[93] Thomas, J. W., Numerical partial differential equations: finite difference methods,

vol. 22 of Texts in Applied Mathematics, Springer-Verlag, New York, 1995.

[94] Trefethen, L. N., Group velocity in finite difference schemes, SIAM Rev., 24 (2),

113–136 (1982).

[95] Tucsnak, M. and Weiss, G., Passive and conservative linear systems, Preprint.

[96] ——, Simultaneous exact controllability and some applications, SIAM J. Control Op-

tim., 38 (5), 1408–1427 (electronic) (2000).

BIBLIOGRAPHY 157

[97] Vichnevetsky, R. and Bowles, J. B., Fourier analysis of numerical approximations of

hyperbolic equations, vol. 5 of SIAM Studies in Applied Mathematics, Society for

Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1982.

[98] Weiss, G., Admissible observation operators for linear semigroups, Israel J. Math.,

65 (1), 17–43 (1989).

[99] Weiss, G., Staffans, O. J. and Tucsnak, M., Well-posed linear systems—a survey with

emphasis on conservative systems, Int. J. Appl. Math. Comput. Sci., 11 (1), 7–33

(2001), mathematical theory of networks and systems (Perpignan, 2000).

[100] Young, R. M., An introduction to nonharmonic Fourier series, vol. 93 of Pure and

Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers],

New York, 1980.

[101] ——, An introduction to nonharmonic Fourier series, Academic Press Inc., San Diego,

CA, 2001, 1st ed.

[102] Zhang, X., Explicit observability estimate for the wave equation with potential and its

application, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (1997), 1101–1115

(2000).

[103] ——, A remark on null exact controllability of the heat equation, SIAM J. Control

Optim., 40 (1), 39–53 (electronic) (2001).

[104] Zhang, X., Zheng, C. and Zuazua, E., Exact controllability of the time discrete wave

equation, Preprint on Discrete Contin. Dyn. Syst.

[105] Zheng, C., Controllability of time discrete heat equation, Submitted to Asymptot.

Anal.

[106] ——, Uniform controllability of the 1-d semi-discrete heat equation, DEA, Universi-

dad Autonoma de Madrid, Spain, 2005.

[107] Zuazua, E., Some problems and results on the controllability of partial differential

equations, in European Congress of Mathematics, Vol. II (Budapest, 1996), vol. 169

of Progr. Math., pp. 276–311, Birkhauser, Basel, 1998.

[108] ——, Boundary observability for the finite-difference space semi-discretizations of the

2-D wave equation in the square, J. Math. Pures Appl. (9), 78 (5), 523–563 (1999).

158 BIBLIOGRAPHY

[109] ——, Controllability of partial differential equations and its semi-discrete approxima-

tions, Discrete Contin. Dyn. Syst., 8 (2), 469–513 (2002).

[110] ——, Propagation, observation, and control of waves approximated by finite difference

methods, SIAM Rev., 47 (2), 197–243 (electronic) (2005).

[111] ——, Control and numerical approximation of the wave and heat equations, in In-

ternational Congress of Mathematicians. Vol. III, pp. 1389–1417, Eur. Math. Soc.,

Zurich, 2006.

[112] ——, Controllability and observability of partial differential equations: Some results

and open problems, in Evolutionary Differential Equations, vol. 8, pp. 527–621, Else-

vier Science, 2006.

departamento de matematicas´math0.bnu.edu.cn/~zhengc/material/thesis.pdf · 2015. 3. 9. ·...

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