viscosity solutions of dynamic-programming equations … · viscosity solutions of...

Arch. Rational Mech. Anal. 163 (2002) 295–327Digital Object Identifier (DOI) 10.1007/s002050200203

Viscosity Solutions of Dynamic-ProgrammingEquations for the Optimal Control of the

Two-Dimensional Navier-Stokes Equations

Fausto Gozzi, S. S. Sritharan & Andrzej Swiech

Communicated by L. C. Evans

Abstract

In this paper we study the infinite-dimensional Hamilton-Jacobi equation as-sociated with the optimal feedback control of viscous hydrodynamics. We resolvethe global unique solvability problem of this equation by showing that the valuefunction is the unique viscosity solution.

1. Introduction

Optimal control of Navier-Stokes equations has many applications in traditionalaerodynamic and hydrodynamic drag reduction, lift enhancement, turbulence con-trol and combustion control. Data assimilation in meteorology and oceanographyinvolves the task of finding best initial data and unknown boundary and body forcesby optimal-control methodology [40]. During the past decade several fundamentaladvances have been made in developing an organized mathematical theory of thissubject [16,19,32]. One of the open problems identified in [33] was the globalunique solvability of the infinite-dimensional Hamilton-Jacobi equation associatedwith the feedback synthesis for Navier-Stokes equations. The problem is resolvedin this paper. In this respect, this paper is at the crossroads of two theoretical devel-opments which have taken place in the past several years, namely flow-control the-ory and the theory of viscosity solutions for infinite-dimensional Hamilton-Jacobiequations. Local solvability for the Hamilton-Jacobi equation associated with thefeedback control of Navier-Stokes equation has been achieved in the past in thecontext ofH∞-control theory [4] and time-periodic control [1]. Global solvabilityfor mild solutions (in the semigroup formulation) of a semilinear Hamilton-Jacobiequation associated with the stochastic control of two-dimensional Navier-Stokesequations is given in [12] where the existence of smooth solutions is proved in some

296 Fausto Gozzi, S. S. Sritharan & Andrzej Swiech

cases. For the impulse-control problem, the Hamilton-Jacobi equation becomes aquasi-variational inequality and this is again resolved in a mild sense in [26]. Theviscosity-solution method which provides another notion of a generalized solutionwas first introduced in the context of finite-dimensional Hamilton-Jacobi equationsbyCrandall & Lions [9] and was generalized to infinite dimensions by a numberof authors. Of particular relevance to this paper are [10, Parts IV, V, VII] and [7,17,18,22]. For other literature on infinite-dimensional Hamilton-Jacobi equations werefer the reader to [2,3,6,23,25,35,36], [10, Part VI] and the references therein.This paper deals with Hamilton-Jacobi-Bellman (HJB) equations associated withoptimal control of Navier-Stokes equations in two-dimensional domains under ho-mogeneous boundary conditions. Periodic boundary conditions can also be treatedby our methods but we do not do it here as this case is easier.

The approach presented here allows natural extensions to second order HJBequations of this type. We think that modifications of the methods developed herecan be applied to prove comparison results for such equations. However, connectingthese second order HJB equations to optimal control of stochastic Navier-Stokesequations seems to be a challenging and difficult problem (see [12]).

Finally we mention that after this paper had been submitted we learned abouta manuscript [31] that deals with abstract infinite-dimensional Hamilton-Jacobiequations and uses a different notion of viscosity solution. The results of [31] canbe applied to some HJB equations coming from the optimal control of Navier-Stokesequations.

In the rest of this section we will describe the mathematical problem. In Section 2we establish various continuous-dependence theorems for the controlled Navier-Stokes equation that are needed in the paper. In Section 3 we describe a suitablenotion of the viscosity solution used in this paper. In Section 4 we prove the unique-ness theorem for these viscosity solutions for time-independent and time-dependentHamilton-Jacobi equations. Section 5 is devoted to the application of these abstractPDE (partial differential equation) results to an optimal-control problem. In Sec-tion 5.1 we show that the value function has the required continuity properties, andin Section 5.2 we prove that it is indeed a viscosity solution in the sense of thedefinition given in Section 3. In Section 5.3 we briefly discuss the applicability ofour results.

Let us now describe a typical optimal-control problem for the abstract Navier-Stokes equation [32, Chapter I]. We will assume throughout that the the kinematicviscosity is equal to 1. Let� ⊂ IR2 be an open and bounded set with smoothboundary. Let

H = the closure of{

x ∈ D(�; IR2

), div x = 0

}in L2

(�; IR2

)V = the closure of

{x ∈ D

(�; IR2

), div x = 0

}in H 1

(�; IR2

),

and letPH be the orthogonal projection inL2(�; IR2

)onto H. Define Ax =

−PH �x, andB(x, y) = PH [(x · ∇)y]. Let t � 0 be the initial time andT � t bethe terminal time. Our state variable is the velocity fieldX : [t, T ] × � → IR2 that

Optimal Control of the Two-Dimensional Navier-Stokes Equations 297

satisfies the equation

dX(s)

ds= −AX(s) − B (X(s), X(s)) + f (s, a(s)) in (t, T ] × H,

X(t) = x ∈ H,(1.1)

wheref : [0, T ] × → V′, is a complete metric space (the control set), anda(·) : [0, T ] → is a measurable function which plays the role of a controlstrategy. We will denote the set of control strategies byU . The minimization, overall controlsa(·) ∈ U , of a cost functional

J (t, x; a) =∫ T

t

l(s, X(s), a(s))ds + g(X(T ))

formally leads to the Hamilton-Jacobi-Bellman (HJB) partial differential equationin (0, T ) × H

ut −〈Ax + B(x, x), Du〉+ inf a∈ {〈f(t, a), Du〉 + l(t, x, a)} = 0 in (0, T ) × H,

u(T , x) = g(x) in H.

(1.2)

This equation should be satisfied by the value function

V(t, x) = infa(·)∈U

J (t, x; a).

Equation (1.2) falls into a general class of infinite-dimensional Hamilton-Jacobiequations

ut − 〈Ax + B(x, x), Du〉 + F(t, x, Du) = 0 in (0, T ) × Hu(T , x) = g(x) in H

(1.3)

that will be the object of our study. Likewise we will be investigating stationaryversions of (1.3),

λu + 〈Ax + B(x, x), Du〉 + F(x, Du) = 0 in H, (1.4)

that are related to the infinite-horizon optimal-control problems.

2. Preliminaries on the two-dimensional controlled Navier-Stokes equation

2.1. Abstract spaces and the Stokes operator

Throughout the paper� will be an open and bounded subset of IR2 with smoothboundary. We denote byWm,p

(�; IR2

)(or simply byWm,p (�) ) the Sobolev

space of order 0� m ∈ IR and powerp � 1 of functions with values in IR2 (which

can be seen as a product spaceWm,p(�; IR2

) = [Wm,p (�; IR)

]2). The norm ofx ∈ Wm,p

(�; IR2

)will be denoted by‖x‖m,p. We will use the notationLp for


W0,p. Moreover, whenp = 2, we will write Hm for Wm,2. We will also be usingthe negative Sobolev spacesH−m. The spaceH can be alternatively defined by

H ={

x ∈ L2(�; IR2

), div x = 0 in D′ (�; IR2

), x · n = 0 in H− 1

2 (∂�)},

wheren is the outward normal to the boundary, see [37, Chapter 1]. The operator

A = −PH �

with the domain

D (A) = H 2(�; IR2

)∩ V

is called the Stokes operator. It is well known thatA is positive definite, self-adjoint,A−1 is compact, andA generates an analytic semigroup. Forγ � 0 we denote byVγ the domain ofA

γ2 , D(A

γ2 ), equipped with the norm

‖x‖γ = ‖Aγ2 x‖0,2. (2.5)

Forγ < 0 the spaceVγ is defined as the completion ofH under the norm (2.5). Ifγ > −1

2 the norm ofVγ is equivalent to the norm ofHγ (see [16, Lemma 4.5]).Moreover the spaceV1 coincides withV. Identifying H with its dual, the spaceV−γ is the dual ofVγ for γ > 0. We will also be using the customary notationV′for the dual ofV and the duality pairing betweenV′ andV will be denoted by〈·, ·〉.The same symbol will also be used to denote the inner product inH if both entriesare inH. Finally the operatorsAγ are isometries betweenV2δ = andV2(δ−γ ) foreach realγ andδ. We recall below Sobolev imbeddings and other inequalities that

we will need in the remainder of the paper.

– Sobolev-imbedding type inequalities: If m � 0, mp � 2 andp � q � 2p2−mp

thenWm,p (�) ↪→ Lq (�), i.e. ,

‖x‖0,q � C ‖x‖m,p for x ∈ Wm,p (�) ,

(noting that whenmp = 2 the imbedding holds for allq < +∞). Combining theabove with the equivalence of norms ofVγ andHγ , we find that forγ ∈ (0, 1]

andq ∈[2, 2

1−γ

](q ∈ [2, +∞) if γ = 1) Vγ ↪→ Lq (�), i.e.,

‖x‖0,q � C ‖x‖γ for x ∈ Vγ . (2.6)

– Gagliardo-Nirenberg type inequality:

‖x‖0,4 � C ‖x‖120 ‖x‖

121 for x ∈ V. (2.7)

– Interpolation inequality: Recall that if an operatorS generates an analytic semi-group, then there exists a constantC such that, for everyx ∈ D(S) and 0� γ �1,

‖Sγ z‖0 � C‖Sz‖γ0‖z‖1−γ

0 . (2.8)


2.2. The Euler operator (inertia term) B

We define the trilinear formb(·, ·, ·) : V × V × V → IR as

b (x, y, z) =∫

�

z(ξ) · (x(ξ) · ∇ξ )y(ξ) dξ

and the bilinear operatorB(·, ·) : V × V → V′ as

〈B(x, y), z〉 = b (x, y, z)

for all z ∈ V. This is just another way to introduce the operatorB that we havealready used in the Navier-Stokes equation (1.1). We will writeB (x) for B (x, x)

when no confusion arises. This operator can be extended in different topologies.By the incompressibility condition we have

b (x, y, y) = 0, b (x, y, z) = −b (x, z, y) . (2.9)

Moreover we have the following estimates forb(x, y, z), with x, y, z ∈ V:

1. By Holder inequality, forp, q > 1, 1p

+ 1q

+ 12 = 1,

|b (x, y, z)| � C ‖x‖0,p ‖y‖1 ‖z‖0,q . (2.10)

2. By (2.7) we obtain

|b (x, y, z)| � C ‖x‖0,4 ‖y‖1 ‖z‖0,4 � C ‖x‖120 ‖x‖

121 ‖y‖1 ‖z‖

120 ‖z‖

121 (2.11)

which gives, whenx = z,

|b (x, y, x)| � C ‖x‖20,4 ‖y‖1 � C ‖x‖0 ‖x‖1 ‖y‖1 . (2.12)

2.3. The controlled Navier-Stokes equation: existence, uniqueness and continuousdependence

In this section we derive various estimates for solutions of the Navier-Stokesequation (1.1). We begin with the definition of solution (see for instance [37,38]).

Definition 1. (i) A weak solution of (1.1) is a map

X ∈ C([t, T ]; H) ∩ L2(t, T ; V) with X′ ∈ L2(t, T ; V−1)

such that (1.1) is satisfied as an equality inV−1, i.e., for everyz ∈ V and almosteverys ∈ (t, T ), we have⟨

X′(s), z⟩ = 〈−AX(s) − B(X(s), X(s)) + f (s, a(s)) , z〉 .

(ii) A strong solution of (1.1) is a map

X ∈ C([t, T ]; V) ∩ L2(t, T ; D(A)) with X′ ∈ L2(t, T ; H)

such that (1.1) is satisfied as an equality inH for almost everys ∈ (t, T ).


Given 0� t � T , an admissible control strategya(·) ∈ U and an initial datumx ∈ H, a weak (or strong) solution of the state equation (1.1) will be denoted byX(·; t, x, a). We will often denote it simply byX(·) when there is no possibility ofconfusion. We first recall the standard existence and uniqueness results [37] for theNavier-Stokes equation (1.1).

Theorem 1. Let f : [0, T ]× → V−1 be bounded and continuous. Let 0 � t � T ,and let a(·) ∈ U .

(i) Given an initial datum x ∈ H, there exists a unique weak solution X(·; t, x, a)

of the state equation (1.1). Moreover, the following estimate holds:

supt�s�T

‖X(s)‖20 +

∫ T

t

‖X(s)‖21ds � ‖x‖2

0 +∫ T

t

‖f(s, a(s))‖2−1 ds. (2.13)

(ii) Given an initial datum x ∈ V and assuming moreover that f : [0, T ] × → His bounded and continuous, there exists a unique strong solution X(·; t, x, a)

of (1.1). Moreover, the following estimate holds:

supt�s�T

‖X(s)‖21 +

∫ T

t

‖AX(s)‖20ds

� ‖x‖21 exp(E(t, T )) +

∫ T

t

‖f(s, a(s))‖20 exp(E(s, T )) ds (2.14)

where

E(t, τ ) := C

∫ τ

t

‖X(r)‖20‖X(r)‖2

1dr � C supt�r�τ

‖X(r)‖20

∫ τ

t

‖X(s)‖21ds.

The next proposition gathers continuous-dependence estimates for solutions ofequation (1.1). The negative norm estimates given here generalize the results givenin [32, Chapter I]).

Proposition 1. Let f : [0, T ] × → V−1 be bounded and continuous. Then:

(i) Let 12 � α � 1. There exist constants C > 0, depending only on the indicated

variables and independent of a(·) ∈ U, such that for all initial conditionsx, y ∈ H,

‖X(s) − Y (s)‖−α

� C

(||x||0 , ||y||0 , sup

a(·)∈U

∫ T

t

‖f(r, a(r))‖2−1 dr

)‖x − y‖−α, (2.15)

∫ T

t

‖X(s) − Y (s)‖21−α

� C

(||x||0 , ||y||0 , sup

a(·)∈U

∫ T

t

‖f(r, a(r))‖2−1 dr

)‖x − y‖2−α, (2.16)


||X(s) − x||−α

� C

(||x||0 , sup

a(·)∈U

∫ T

t

‖f(r, a(r))‖2−1 dr

)(s − t)

14 , (2.17)

and there is a modulus σ , independent of the strategies a(·) ∈ U, such that

||X(s) − x||0 � σ(s − t). (2.18)

(ii) For every initial condition x ∈ V there exists a constant C > 0 independent ofthe strategies a(·) ∈ U such that

||X(s) − x||0 � C

(||x||1 , sup

s∈[t,T ],a∈

‖f (s, a)‖2−1

)(s − t)

12 , (2.19)

and if in addition f : [0, T ] × → H is bounded and continuous, then

||X(s) − x||1 � σ(s − t) (2.20)

for some modulus σ independent of the strategies a(·) ∈ U .

Proof. (i) We first prove (2.15). LetX(s) andY (s) be the solutions of (1.1) withinitial conditionsx andy respectively. LetZ(s) = X(s) − Y (s). Then taking theinner products of (1.1) forX(s) andY (s) with A−αZ(s) we get, fors ∈ [t, T ],⟨

X′(s), A−αZ (s)⟩ = ⟨−AX(s) − B(X(s)) + f (s, a(s)) , A−αZ (s)

⟩,

X(t) = x,(2.21)

⟨Y ′(s), A−αZ (s)

⟩ = ⟨−AY (s) − B(Y (s)) + f (s, a(s)) , A−αZ (s)⟩,

Y (t) = y.(2.22)

Subtracting the resulting equalities, we obtain⟨Z′(s), A−αZ (s)

⟩ = − ⟨AZ(s), A−αZ (s)⟩+ ⟨

B(Y (s)) − B(X(s)), A−αZ (s)⟩,

Z(t) = x − y,

(2.23)

so

1

2

d

ds‖A− α

2 Z(s)‖20 + ‖A

1−α2 Z(s)‖2

0

= −b(X(s), X(s), A−αZ (s)

)+ b(Y (s), Y (s), A−αZ (s)

). (2.24)

Now, by linearity, adding and subtractingb(Y (s), X(s), A−αZ (s)

),

− b(X(s), X(s), A−αZ (s)

)+ b(Y (s), Y (s), A−αZ (s)

)= −b(Z(s), X(s), A−αZ(s)) − b(Y (s), Z(s), A−αZ(s)).


By Holder inequality we have (see (2.10))

|b(Z(s), X(s), A−αZ(s))| � C‖Z(s)‖0, 2α‖X(s)‖1‖A−αZ(s)‖0, 2

1−α.

Sobolev imbeddings give us (see (2.6))

‖Z(s)‖0, 2α

� C‖Z(s)‖1−α

and

‖A−αZ(s)‖0, 21−α

� C ‖A−αZ(s)‖α = C‖Z(s)‖−α.

Therefore we obtain

|b(Z(s), X(s), A−αZ(s))| � C‖Z(s)‖1−α‖X(s)‖1‖Z(s)‖−α

� 14‖Z(s)‖2

1−α + C‖X(s)‖21‖Z(s)‖2−α (2.25)

for some constantC > 0. The second term is a little more difficult to estimate.Similarly, we have

|b(Y (s), Z(s), A−αZ(s))| = |b(Y (s), A−αZ(s), Z(s))|� C‖Y (s)‖0, 2

1−α‖A−αZ(s)‖1‖Z(s)‖0, 2

α

= C‖Y (s)‖0, 21−α

‖A12−αZ(s)‖0‖Z(s)‖0, 2

α.

Estimates (2.6) and (2.8) give

‖Y (s)‖0, 21−α

� C‖Aα2 Y (s)‖0 = C‖(A

12 )αY (s)‖0 � C‖Y (s)‖α

1‖Y (s)‖1−α0 ,

‖A12−αZ(s)‖0 = ‖(A

12 )1−α(A− α

2 Z(s))‖0 � C‖A1−α

2 Z(s)‖1−α0 ‖A− α

2 Z(s)‖α0,

‖Z(s)‖0, 2α

� C‖A1−α

2 Z(s)‖0,

and, by (2.13),

sups∈[t,T ]

‖Y (s)‖1−α0 �

(‖y‖2

0 +∫ T

t

‖f(s, , a(s))‖2−1 ds

) 1−α2

.

Therefore collecting these estimates and using Young’s inequality we obtain

|b(Y (s), Z(s), A−αZ(s))|� C‖Y (s)‖α

1‖Y (s)‖1−α0 ‖A

1−α2 Z(s)‖2−α

0 ‖A− α2 Z(s)‖α

0

� C‖Y (s)‖α1‖A

1−α2 Z(s)‖2−α

0 ‖A− α2 Z(s)‖α

0 (2.26)

� 14‖Z(s)‖2

1−α + C‖Y (s)‖21‖Z(s)‖2−α

Using (2.25) and (2.26) in (2.24) produces

1

2

d

ds‖Z(s)‖2−α + 1

2‖A

1−α2 Z(s)‖2 � C(‖X(s)‖2

1 + ‖Y (s)‖21)‖Z(s)‖2−α. (2.27)


Gronwall’s inequality and (2.14) now yield

‖X(s) − Y (s)‖2−α = ‖Z(s)‖2−α � eC∫ st (‖X(r)‖2

1+‖Y (r)‖21)dr‖x − y‖2−α

� C

(||x||0 , ||y||0 ,

∫ T

t

‖f(r, a(r))‖2−1 dr

)‖x − y‖2−α

(2.28)

which gives us (2.15). Then putting (2.28) into (2.27) we get (2.16). To prove (2.17)we setY (s) = X (s) − x so that

Y ′ (s) = X′ (s) = −AX (s) − B (X (s)) + f (s, a (s)) . (2.29)

Taking the inner product of (2.29) withA−αY (s), we get

1

2

d

ds‖Y (s)‖2−α = −

⟨A1−αX(s), Y (s)

⟩(2.30)

−b(X (s) , X (s) , A−αY (s)

)+ ⟨f (s, a (s)) , A−αY (s)

⟩.

Sinceα � 12 we have, using also (2.13),

|〈A1−αX(s), Y (s)〉| � ‖A1−αX (s) ‖0 ‖Y (s)‖0 � C‖X(s)‖1.

Moreover by (2.12) and (2.13)∣∣b (X (s) , X (s) , A−αY (s))∣∣ � C ‖X (s)‖0 ‖A−αY (s)‖1 ‖X (s)‖1

� C‖X(s)‖1‖Y (s)‖0‖X(s)‖0 � C‖X(s)‖1.

Finally∣∣⟨f (s, a (s)) , A−αY (s)⟩∣∣ � ‖f (s, a (s))‖−1

∥∥A−αY (s)∥∥

1

� C ‖f (s, a (s))‖−1 ‖Y (s)‖0 � C ‖f (s, a (s))‖−1 .

Putting these inequalities in (2.30) produces

1

2

d

ds‖Y (s)‖2−α � C

(‖X(s)‖1 + ‖f (s, a (s))‖−1).

Integrating and using the Cauchy-Schwarz inequality, we then get

‖X(s) − x‖2−α

� C

(∫ s

t

‖X(r)‖21dr

) 12 +

(∫ T

t

‖f (r, a (r)) ‖2−1ds

) 12

(s − t)

12

and the claim follows. To prove the fourth estimate we argue by contradiction. Wewant to show that

sups∈[t,t+ε],a(·)∈U

‖X(s; t, x, a) − x‖0 → 0 asε → 0.


Assume that this is false. Then there exist sequencessn, an(·) such that

sn → t, ‖X(sn; t, x, an) − x‖0 � ε.

However, by the boundedness of‖X(sn; t, x, an)‖0 and the third estimate of (i) wehave

X(sn; t, x, an) → x strongly inV−α and weakly inH.

Moreover, weak convergence inH implies that

lim infn→∞ ‖X(sn; t, x, an)‖0 � ‖x‖0,

and from the energy inequality (2.13) we can conclude that

lim supn→∞

‖X(sn; t, x, an)‖0 � ‖x‖0.

Combining these two results gives convergence of the norms and this, along withthe weak convergence inH, yieldsX(sn; t, x, an) → x strongly inH , which is acontradiction.

(ii) Let X (s) = X (s; t, x, a) be the weak solution of (1.1) with initial conditionsx at timet . Set as beforeY (s) = X (s) − x so thatY (s) satisfies (2.29). Taking theinner product of (2.29) withY (s) we obtain

1

2

d

ds‖Y (s)‖2

0 = − 〈AX(s), Y (s)〉 − b (X (s) , X (s) , Y (s))

+ 〈f (s, a (s)) , Y (s)〉 (2.31)

= − 〈AY (s), Y (s)〉 − 〈Ax, Y (s)〉−b (X (s) , X (s) , Y (s)) + 〈f (s, a (s)) , Y (s)〉 .

Now by the tri-linearity ofb(·, ·, ·) and the orthogonality relations (2.9)

b (X (s) , X (s) , Y (s)) = b (Y (s) , X (s) , Y (s)) + b (x, X (s) , Y (s))

= b (Y (s) , X (s) , Y (s)) + b (x, x, Y (s)) ,

which yields, when plugged into (2.31),

1

2

d

ds‖Y (s)‖2

0 + ‖Y (s)‖21 = − 〈Ax, Y (s)〉 − b (Y (s) , X (s) , Y (s)) (2.32)

+b (x, x, Y (s)) + 〈f (s, a (s)) , Y (s)〉 .

Now we have

|〈Ax, Y (s)〉| =∣∣∣⟨A 1

2 x, A12 Y (s)

⟩∣∣∣ � ‖x‖1 ‖Y (s)‖1 � 18 ‖Y (s)‖2

1 + C. (2.33)

Moreover, by (2.12),

|b (Y (s) , X (s) , Y (s))| � C ‖Y (s)‖0 ‖Y (s)‖1 ‖X (s)‖1

� 18 ‖Y (s)‖2

1 + C ‖Y (s)‖20 ‖X (s)‖2

1 (2.34)


and

|b (x, x, Y (s))| � C ‖x‖0 ‖x‖1 ‖Y (s)‖1 � 18 ‖Y (s)‖2

1 + C, (2.35)

and finally

|〈f (s, a (s)) , Y (s)〉| � ‖f (s, a (s))‖−1 ‖Y (s)‖1

� 18 ‖Y (s)‖2

1 + C ‖f (s, a (s))‖2−1 . (2.36)

Using (2.33)–(2.36) in (2.32) produces

1

2

d

ds‖Y (s)‖2

0 + 1

2‖Y (s)‖2

1 � C[1 + ‖Y (s)‖2

0 ‖X (s)‖21 + ‖f (s, a (s))‖2−1

].

Therefore it follows by Gronwall’s inequality that

‖X (s) − x‖20 � C

[(s − t) +

∫ s

t

‖f (r, a (r))‖2−1 dr

]eC

∫ st‖X(r)‖2

1dr ,

from which our claim (2.19) easily follows. Moreover we note that, settingω1 (r − t) = ‖X (s) − x‖2

0, we also have∫ s

t

‖X(r) − x‖21dr � C

[(s − t) +

∫ s

t

‖f (r, a (r))‖2−1 dr

]

+C

∫ s

t

ω1 (r − t) ‖X (r)‖21 dr.

The proof of (2.20) is the same as the proof of estimate (2.18) in part (i) once weknow that lim sups→t ‖X (s) ‖0 � ‖x‖0, which follows from the assumption thatf is bounded inH and from (2.14). ��

3. Viscosity solutions of the HJB equation

The definition of a viscosity solution that we propose here has its predecessorsin [10] and [7,17,18]. We just have to use appropriate test functions.

Definition 2. A functionψ is atest function for equation (1.3) ifψ = ϕ+δ(t) ||x||20,where

• ϕ ∈ C1 ((0, T ) × H), A12 Dϕ is continuous, andϕ is weakly sequentially lower-

semicontinuous,• δ ∈ C1 ((0, T )) is such thatδ > 0 on(0, T ).

Definition 3. A functionψ is atest function for equation (1.4) ifψ = ϕ + δ ||x||20,where

• ϕ ∈ C1 (H), A12 Dϕ is continuous, andϕ is weakly sequentially lower-semiconti-

nuous,• δ > 0.


In the definition below we assume thatF : [0, T ] × V × V → IR.

Definition 4. A functionu ∈ C ((0, T ) × H) is aviscosity subsolution (supersolu-tion) of (1.3) if for every test functionψ , wheneveru − ψ has a local maximum(respectivelyu + ψ has a local minimum) at(t, x) thenx ∈ V and

ψt(t, x) − 〈Ax + B(x, x), Dψ(t, x)〉 + F(t, x, Dψ(t, x)) � 0

(respectively

− (ψt (t, x) − 〈Ax + B(x, x), Dψ(t, x)〉) + F(t, x, −Dψ(t, x)) � 0.)

A function is aviscosity solution if it is both a viscosity subsolution and a viscositysupersolution.

Definition 5. A function u ∈ C(H) is a viscosity subsolution (supersolution) of(1.4) if for every test functionψ , wheneveru−ψ has a local maximum (respectivelyu + ψ has a local minimum) atx thenx ∈ V and

λu(x) + 〈Ax + B(x, x), Dψ(x)〉 + F(x, Dψ(x)) � 0

(respectively

λu(x) − 〈Ax + B(x, x), Dψ(x)〉 + F(x, −Dψ(x)) � 0.)

A function is aviscosity solution if it is both a viscosity subsolution and a viscositysupersolution.

The somehow arbitrary choice of the function‖ · ‖20 as part of the test function

was made for simplicity, and because in this paper we only deal with linearlygrowing solutions. Other radial functions can be chosen to handle more generalcases.

4. Comparison principles

In this section we prove comparison principles for viscosity solutions that inturn imply the uniqueness of solutions of (1.3) and (1.4) under certain conditions.We will assume the following hypothesis.

Hypothesis 1. The Hamiltonian F : [0, T ] × Vβ × V → IR, and there exist amodulus of continuity ω, a constant β in the range 0 < β < 1

2 , and moduli ωR

such that, for every R > 0,

|F(t, x, p) − F(t, y, p)| � ωR

(‖x − y‖β

)+ ω(‖x − y‖β‖p‖0

)(4.37)

if ‖x‖0, ‖y‖0 � R,

|F(t, x, p) − F(t, x, q)| � ω ((1 + ‖x‖0)‖p − q‖1) , (4.38)

|F(t, x, p) − F(s, x, p)| � ωR (|t − s|) , if ‖x‖0, ‖y‖0, ‖p‖0 � R. (4.39)


Theorem 2. Let Hypothesis 1 hold. Let u, v : (0, T ) × H → IR be respectively aviscosity subsolution, and a viscosity supersolution of (1.3). Let

u(t, x), −v(t, x), |g(x)| � C(1 + ‖x‖0)

and let u(t, ·), v(t, ·), g be locally Lipschitz continuous in ‖·‖β−1 norm on boundedsubsets of H, uniformly for t in closed subsets of (0, T ). Let moreover{

(i) lim t↑T (u(t, x) − g(x))+ = 0(ii ) lim t↑T (v(t, x) − g(x))− = 0

(4.40)

uniformly on bounded subsets of H. Then u � v.

Theorem 3. Let Hypothesis 1 hold. Let u, v : H → IR be respectively a viscositysubsolution, and a viscosity supersolution of (1.4). Let u, −v be bounded fromabove and be Lipschitz continuous in ‖ · ‖β−1 norm on bounded subsets of H. Thenu � v.

Proof of Theorem 3. Setα = 1 − β and assume for simplicity thatλ = 1. Forε, δ > 0 we consider the function

7(x, y, ε, δ) := u(x) − v(y) − ‖x−y‖2−α

2ε− δ(‖x‖2

0 + ‖y‖20).

Because of the continuity assumptions onu, v and the fact thatA−1 is compact,the function above is weakly sequentially upper-semicontinuous and so it attains aglobal maximum at some pointsx, y ∈ V (by the definition of viscosity solution).For a fixedδ we will show (as for instance in [22]) that the pointsx and y arebounded independently ofε,

lim supδ→0

lim supε→0

δ(‖x‖20 + ‖y‖2

0) = 0, (4.41)

and

lim supε→0

‖x − y‖2−α

2ε= 0 for fixed δ. (4.42)

(Please notice that we have reversed the order in which we pass to the limits in theabove expressions compared with [22].) To see this we set

m1(ε, δ) = supx,y∈H

7(x, y, ε, δ),

m2(δ) = supx,y∈H

{u(x) − v(y) − δ

(‖x‖2

0 + ‖y‖20

)}

and note that we have

m = limδ↓0

m2(δ) and m2(δ) = limε↓0

m1(ε, δ).


Now,

m1(ε, δ) = 7(x, y, ε, δ) = u(x) − v(y) − ‖x−y‖2−α

2ε− δ(‖x‖2

0 + ‖y‖20)

and, for fixedδ,

m1(ε, δ) + ‖x−y‖2−α

4ε= u(x) − v(y) − ‖x−y‖2−α

4ε− δ(‖x‖2

0 + ‖y‖20)

� m1(2ε, δ).

Thus

‖x−y‖2−α

4ε�m1(2ε, δ) − m1(ε, δ).

This gives (4.42). Similarly,

m1(ε, δ) + δ

2(‖x‖2

0 + ‖y‖20) = u(x) − v(y) − ‖x−y‖2−α

2ε− δ

2(‖x‖2

0 + ‖y‖20)

� m1

(ε,

δ

2

)which gives

δ

2(‖x‖2

0 + ‖y‖20) � m1(ε, δ) − m1

(ε,

δ

2

),

from which we obtain (4.41). Using the definition of viscosity solution andb(x, x, x) = 0, we now have

u(x) + 2δ〈Ax, x〉 + 1

ε〈Ax, A−α(x − y)〉

+ 1

εb(x, x, A−α(x − y)) + F

(x,

1

εA−α(x − y) + 2δx

)� 0

and

v(y) − 2δ〈Ay, y〉 + 1

ε〈Ay, A−α(x − y)〉

+ 1

εb(y, y, A−α(x − y)) + F

(y,

1

εA−α(x − y) − 2δy

)� 0.

Combining these two inequalities we obtain

u(x) − v(y) + 2δ(‖x‖21 + ‖y‖2

1) + 1

ε‖x − y‖2

1−α

+ 1

ε(b(x, x, A−α(x − y)) − b(y, y, A−α(x − y)))

+ F(

x,1

εA−α(x − y) + 2δx

)− F

(y,

1

εA−α(x − y) − 2δy

)� 0. (4.43)


We are now going to estimate the crucial second line in (4.43). We have

b(x, x, A−α(x − y)) − b(y, y, A−α(x − y))

= b(x − y, x, A−α(x − y)) + b(y, x − y, A−α(x − y)).

Both terms above can be estimated similarly and so we will only show how toestimate the first one. We have, by H¨older inequality,

1

ε|b(x − y, x, A−α(x − y))|

= 1

ε|b(x − y, A−α(x − y), x)|

� c

ε‖x‖0,q‖A−α+ 1

2 (x − y)‖0,2p‖x − y‖0,2p, (4.44)

where we tookp andq such that1p

+ 1q

= 1, and laterp will be sufficiently small.

To continue we will use inequality (2.8). We chooseτ such that 0< τ < α − 12.

To estimate‖A−α+ 12 (x − y)‖0,2p we first notice that ifp is sufficiently close to 1,

Sobolev imbedding (2.6) guarantees that

‖A−α+ 12 (x − y)‖0,2p � C‖A−α+ 1

2+ τ2 (x − y)‖0.

We now setS = A12 , andz = A− α

2 (x − y) in (2.8). Then

‖A−α+ 12+ τ

2 (x − y)‖0 = ‖S1−α+τ z‖0

� C‖Sz‖1−α+τ0 ‖z‖α−τ

0

= C‖A1−α

2 (x − y)‖1−α+τ0 ‖x − y‖α−τ−α .

To estimate‖x − y‖0,2p we again use the Sobolev imbedding

‖x − y‖0,2p � C‖x − y‖τ(1−α)

that holds ifp is small enough, and then setS = A1−α

2 andz = x − y in (2.8). Wethen obtain

‖x − y‖τ(1−α) = ‖Sτ z‖0 � C‖Sz‖τ0‖z‖1−τ

0 = C‖A1−α

2 (x − y)‖τ0‖x − y‖1−τ

0 .

(4.45)

Therefore, plugging (4.45) and (4.45) into (4.44), and estimating further

‖x‖0,q � C‖x‖1

we get

1

ε|b(x − y, x, A−α(x − y))|

� C

ε‖x‖1‖A

1−α2 (x − y)‖1−α+2τ

0 ‖x − y‖α−τ−α ‖x − y‖1−τ0


which, upon using‖x − y‖0 � C‖A1−α

2 (x − y)‖0, yields

1

ε|b(x − y, x, A−α(x − y))| � C‖x‖1

‖x − y‖1−α√ε

‖x − y‖α−τ−α√ε

‖x − y‖1−α+τ0 .

(4.46)

We now notice that

7(x, y, ε, δ) � {7(x, x, ε, δ), 7(y, y, ε, δ)} . (4.47)

We use the fact thatu andv are locally Lipschitz continuous in‖ · ‖−α norm and‖x‖0, ‖y‖0 � Rδ independently ofε for a fixedδ to deduce from (4.47) that

‖x − y‖2−α

ε� Kδ‖x − y‖−α

for someKδ > 0. This implies that

‖x − y‖12−α√

ε�√

Kδ.

Therefore

‖x − y‖α−τ−α√ε

= ‖x − y‖12−α√

ε‖x − y‖α− 1

2−τ

−α �√

Kδ‖x − y‖α− 12−τ

−α → 0

asε → 0 by (4.42) sinceα − 12 + τ > 0. Using this in (4.46) we thus obtain

1

ε|b(x − y, x, A−α(x − y))| � ‖x‖1

‖x − y‖1−α√ε

σ1(ε, δ)

� δ‖x‖21 + ‖x − y‖2

1−α

2εσ2(ε, δ) (4.48)

for some local moduliσ1 andσ2. Similarly we obtain

1

ε|b(y, x − y, A−α(x − y))| � δ‖y‖2

1 + ‖x − y‖21−α

2εσ2(ε, δ). (4.49)

We can now proceed with estimating (4.43). Using (4.48), (4.49), and Hypothesis1 in (4.43) (notice thatβ = 1 − α) it follows that

u(x) − v(y) + 1

ε‖x − y‖2

1−α − 1

ε‖x − y‖2

1−ασ2(ε, δ) + δ(‖x‖21 + ‖y‖2

1)

− ωRδ (‖x − y‖1−α) − ω

(‖x − y‖1−α

‖x − y‖−α

ε

)� ω(δ(1 + ‖x‖0)‖x‖1) + ω(δ(1 + ‖y‖0)‖y‖1). (4.50)


For fixedµ, δ > 0 we now chooseCµ andCµ,δ so thatω(s) � µ + Cµs andωRδ (s) � µ + Cµ,δs. Then

ω(δ(1 + ‖x‖0)‖x‖1) + ω(δ(1 + ‖y‖0)‖y‖1)

� 2µ + Cµδ ((1 + ‖x‖0)‖x‖1 + (1 + ‖y‖0)‖y‖1)

� 2µ + δ(‖x‖21 + ‖y‖2

1) + Dµδ(1 + ‖x‖2

0 + ‖y‖20

). (4.51)

Moreover,

ωRδ (‖x − y‖1−α) + ω

(‖x − y‖1−α

‖x − y‖−α

ε

)

� 2µ + Cµ‖x − y‖1−α

‖x − y‖−α

ε+ Cµ,δ‖x − y‖1−α

� 2µ + ‖x − y‖21−α

2ε+ Cµ,δ‖x − y‖1−α + 2C2

µ

‖x − y‖2−α

ε. (4.52)

Putting (4.51) and (4.52) into (4.50), and using (4.41) and (4.42), we thereforeobtain

u(x) − v(y) +(

1

2− σ2(ε, δ)

) ‖x − y‖21−α

ε− Cµ,δ‖x − y‖1−α

� 4µ + σ3(ε, δ; µ)

for some functionσ3 such that for every fixedµ

lim supδ→0

lim supε→0

σ3(ε, δ; µ) = 0. (4.53)

We now observe that

limε→0

infr>0

(r2

4ε− Cµ,δr

)= 0. (4.54)

This yields

u(x) − v(y) � 4µ + σ4(ε, δ; µ) (4.55)

for some functionσ4 satisfying (4.53). Finally (4.55) yields

u(x) − v(x) − 2δ‖x‖20 � u(x) − v(y) � 4µ + σ4(ε, δ; µ).

If we now letε → 0, δ → 0 and thenµ → 0, we find thatu � v. ��Proof of Theorem 2. Givenµ > 0, define

uµ(t, x) = u(t, x) − µ

t, vµ(t, x) = v(t, x) + µ

t.

Thenuµ andvµ satisfy respectively

(uµ)t − 〈Ax + B(x, x), Duµ〉 + F(x, Duµ) � µ

T 2


and

(vµ)t − 〈Ax + B(x, x), Dvµ〉 + F(x, Dvµ) � − µ

T 2 .

Let Cµ be such thatω(s) � µ

2T 2 + Cµs, and letKµ = 2C2µ. Forε, δ, γ > 0, and

0 < Tδ < T, we consider the function

uµ(t, x) − vµ(s, y) − ‖x−y‖2−α

2ε− δeKµ(T −t)‖x‖2

0 − δeKµ(T −s)‖y‖20 − (t − s)2

2γ

on (0, Tδ] × H. This function has a global maximum att , s, x, y, where 0< t, s,andx, y are bounded independently ofε for a fixedδ. Moreoverx, y ∈ V. Similarlyto the stationary case we have

lim supδ→0

lim supε→0

lim supγ→0

δ(‖x‖20 + ‖y‖2

0) = 0, (4.56)

lim supε→0

lim supγ→0

‖x − y‖2−α

2ε= 0 for fixed δ, (4.57)

and

lim supγ→0

(t − s)2

2γ= 0 for fixed δ, ε. (4.58)

If u is not less than or equal tov it then follows from (4.57), (4.58) and (4.40) thatfor smallµ andδ, and forTδ sufficiently close toT we havet, s < Tδ if γ andε aresufficiently small. Therefore using the definition of viscosity solution we obtain

t − s

γ− δKµeKµ(T −t )‖x‖2 − 2δeKµ(T −t )〈Ax, x〉 − 1

ε〈Ax, A−α(x − y)〉

− 1

εb(x, x, A−α(x − y)) + F

(t , x,

1

εA−α(x − y) + 2δeKµ(T −t )x

)� µ

T 2

and

t − s

γ+ δKµeKµ(T −s)‖y‖2 + 2δeKµ(T −s)〈Ay, y〉 − 1

ε〈Ay, A−α(x − y)〉

− 1

εb(y, y, A−α(x − y)) + F

(s, y,

1

εA−α(x − y) − 2δeKµ(T −s)y

)� − µ

T 2 .

Combining these two inequalities, we find that

Kµδ(eKµ(T −t )‖x‖2

0 + eKµ(T −s)‖y‖20

)+ 2δ

(eKµ(T −t )‖x‖2

1 + eKµ(T −s)‖y‖21

)+ 1

ε‖x − y‖2

1−α + 1

εb(x, x, A−α(x − y)) − 1

εb(y, y, A−α(x − y))

+ F(s, y,

1


)− F

(t , x,

1


)� − 2

µ

T 2 . (4.59)


We now estimate∣∣∣∣F(t , x,1


)− F

(t , x,

1

εA−α(x − y)

)∣∣∣∣+∣∣∣∣F(s, y,

1


)− F

(s, y,

1

εA−α(x − y)

)∣∣∣∣� ω

((1 + ‖x‖0)2δeKµ(T −t )‖x‖1

)+ ω

((1 + ‖y‖0)2δeKµ(T −s)‖y‖1

)� µ

T 2 + Cµ

((1 + ‖x‖0)2δeKµ(T −t )‖x‖1

)+Cµ

((1 + ‖y‖0)2δeKµ(T −s)‖y‖1

)� µ

T 2 + 2δC2µ

(2 + eKµ(T −t )‖x‖2

0 + eKµ(T −s)‖y‖20

)+δ

(eKµ(T −t )‖x‖2

1 + eKµ(T −s)‖y‖21

).

Using this and the fact thatKµ = 2C2µ in (4.59), we therefore obtain

δ(‖x‖21 + ‖y‖2

1) + 1

ε‖x − y‖2

1−α

+ 1

εb(x, x, A−α(x − y)) − 1

εb(y, y, A−α(x − y))

+ F(s, y,

1

εA−α(x − y)

)− F

(t , x,

1

εA−α(x − y)

)� − µ

T 2 + σ1(δ, µ) (4.60)

for some local modulusσ1. The rest of the proof follows that of the stationary case.We first letγ → 0 and use (4.39) to eliminate the dependence ont and s above,and then estimate all the terms as in the proof of Theorem 3. After doing this weobtain(

1

2− σ2(ε, δ)

) ‖x − y‖21−α

ε− Dµ,δ‖x − y‖1−α � − µ

2T 2 + σ3(γ, ε, δ; µ)

for some constantDµ,δ, a local modulusσ2, and a functionσ3 such that, for everyfixedµ,

lim supδ→0

lim supε→0

lim supγ→0

σ3(γ, ε, δ; µ) = 0.

This inequality, upon employing (4.54), yields a contradiction if we letγ → 0, ε →0 and thenδ → 0. ��

5. Optimal control problem for the Navier-Stokes equation

In this section we study an optimal-control problem for the Navier-Stokes equa-tion. We only consider the finite-horizon problem. Similar results can be obtained


for the infinite-horizon problem by the same methods. We recall that in the finite-horizon case for a fixedT > 0 and 0 � t < T we try to minimize the costfunctional

J (t, x; a (·)) =∫ T

t

l (s, X (s; t, x, a) , a(s)) ds + g (X (T ; t, x, a)), (5.61)

whereX (·; t, x, a) is the solution of (1.1). We assume the following.

Hypothesis 2. There exist a constant 0 < β < 12 , an absolute constant C > 0, and

families of moduli σR and constants LR for R > 0 such that

(i) l is continuous on [0, T ] × Vβ × and, for t, s ∈ [0, T ], x, y ∈ Vβ , anda ∈ ,

|l(t, x, a) − l(s, y, a)| � σR(|t − s|) + LR‖x − y‖β if ‖x‖0, ‖y‖0 � R,

(5.62)

|l (s, x, a)| � C(1 + ‖x‖1); (5.63)

(ii) g : H → IR is such that

|g(x) − g(y)| � LR‖x − y‖β−1 if ‖x‖0, ‖y‖0 � R, (5.64)

|g(x)| � C (1 + ‖x‖0) ; (5.65)

(iii) f : [0, T ] × → H is bounded, continuous, and f(·, a) is uniformly continu-ous, uniformly for a ∈ .

Let us briefly discuss the applicability of our control problem. Note that if wetake as a separable Banach space then the following example fits Hypothesis 2:

f(t, a) = κ(‖a‖ )Na,

whereκ(·) ∈ C∞,0(IR+) here acts like an engineering constraint on the controlactuation, and the control operatorN ∈L( ; H) is a linear continuous operatorrepresenting spatial localization of the control. HereC∞,0(IR+) denotes smoothfunctions which identically vanish outside a finite interval[0, R]. For conductingfluids (even slightly conducting fluids such as salt water) such controllers can berealized through the action of Lorentz force [27, Chapter 10], [8, Chapter IV]. Sincethe Lorentz force is given byj×B wherej is the current andB is the magnetic field,spatial localization of such distributed control forces can be realized by choosinga suitable shape and location for the magnets. This is implemented for example ina boundary-layer-stabilization experiment in [21]. A cost functional of the form

J (t, x; a (·)) =∫ T

t

‖Aβ2 (X(s) − Xd(s))‖0ds

+∫ T

t

λ(‖a(s)‖ )‖a(s)‖2 ds + ‖A

β−12 X(T )‖0


with λ(·) ∈ C∞,0(IR+) being a discount factor andXd(t) either a desired field ora known smooth field satisfies Hypothesis 2. We note that if the control is bounded(control set is bounded) then the discount factorsκ(·) andλ(·) are not needed.Although it is weaker than the enstrophy norm‖A

12 X(s)‖0, the above integrand

‖Aβ2 (X(s) − Xd(s))‖0 still amounts to minimizing spatial turbulence structures

contained in the deviation fieldX(t) − Xd(t) and hence is useful in practice. Indata-assimilation problems in meteorology [11] and oceanography [40] a majorstep is to estimate the unknown distributed forces due to environmental effectswhich cannot be completely incorporated into the model due to the complexityof the actual problem. In these applicationsXd would represent the extrapolatedmeasurements and the unknown time-dependent forces are regarded as controls withpossible spatial localization and hence fit within the above abstract formulation. Theintegrand we chose above is in fact stronger than theL2-norm integrands used inthe literature on data assimilation of geophysics.

We will prove that the value function for the problem,


J (t, x; a (·)) (5.66)

is the unique viscosity solution of the HJB equation

ut − 〈Ax + B(x, x), Du〉+ inf a∈ {〈f(t, a), Du〉 + l(t, x, a)} = 0 in (0, T ) × H,

u(T , x) = g(x) in H.

(5.67)

The lemma below is easy and we omit the proof.

Lemma 1. Under the assumptions of Hypothesis 2 the Hamiltonian F given by

F (t, x, p) = infa∈

{〈f(t, a), p〉 + l(t, x, a)}

is continuous on [0, T ] × Vβ × V and it satisfies (4.37)–(4.39).

5.1. Properties of the value function

The continuity properties of the value function are described by the followingresult.

Proposition 2. Let Hypothesis 2 hold. Let α = 1 − β. Then for every R > 0 thereexists a constant CR such that the value function V defined by (5.66)satisfies

|V(t1, x) − V(t2, y)| � CR

(|t1 − t2| 1

2 + ‖x − y‖−α

)(5.68)

for t1, t2 ∈ [0, T ] and ‖x‖0, ‖y‖0 � R. Moreover

|V(t, x)| � C(1 + ‖x‖0). (5.69)


Proof. We will first prove the local Lipschitz continuity ofV(t, ·). Let

M = MR =(R2 + T K2

) 12

,

whereK is such that supt∈[0,T ],a∈ ‖f(t, a)‖−1 � K. It then follows from (2.13),(2.15) and (2.16) that for‖x‖0, ‖y‖0 � R we have

|V(t, x) − V(t, y)| � supa(·)∈U

|J (t, x; a (·)) − J (t, y; a (·)) |

� supa(·)∈U

(∫ T

t

|l(s, X(s), a(s)) − l(s, Y (s), a(s))|ds

+ |g(X(T )) − g(Y (T ))|)

� LM supa(·)∈U

(∫ T

t

‖X(s) − Y (s)‖β ds + ‖X(T ) − Y (T )‖−α

)

� CR‖x − y‖−α.

Estimate (5.69) is obvious from (2.13), (5.63), and (5.65). It remains to prove thelocal Holder continuity ofV in t . Let 0� t1 � t2 � T . Then

|V(t1, x) − V(t2, x)| � supa(·)∈U

|J (t1, x; a (·)) − J (t2, x; a (·)) |

� supa(·)∈U

∫ t2

t1

|l (s, X (s; t1, x, a(·)) , a(s)) |ds

+ supa(·)∈U

∫ T

t2

|l (s, X (s; t1, x, a(·)) , a(s))

− l(s, X (s; t2, x, a(·)) , a(s))|ds

+ supa(·)∈U

|g (X (T ; t1, x, a(·))) − g (X (T ; t2, x, a(·))) |

so that, writingX1 (s) for X (s; t1, x, a(·)) andX2 (s) for X (s; t2, x, a(·)) , andusing Hypothesis 2 and (2.13),

|V(t1, x) − V(t2, x)| � supa(·)∈U

∫ t2

t1

|l (s, X1(s), a(s))| ds

+ C(‖x‖0) supa(·)∈U

(∫ T

t2

||X1 (s) − X2 (s)||β ds

+ ||X1 (T ) − X2 (T )||−α

).


We now observe that, for everya(·) ∈ U ,∫ t2

t1

|l (s, X1(s), a(s)) |ds � C

∫ t2

t1

(1 + ||X1(s)||1) ds

� C (t2 − t1)12

(∫ t2

t1

(1 + ||X1(s)||21

)ds

) 12

� C(‖x‖0) (t2 − t1)12 ,

and since

X1 (s) = X (s; t1, x, a(·)) = X (s; t2, X(t2; t1, x, a(·)), a(·)) ,

we have by (2.15), (2.16) and (2.17)

supa(·)∈U

(∫ T

t2

||X1 (s) − X2 (s)||2β ds + ||X1 (T ) − X2 (T )||2−α

)

� C(‖x‖0)|(t2 − t1)12 .

Combining the last two inequalities we therefore obtain

|V(t1, x) − V(t2, x)| � C(‖x‖0)(t2 − t1)12 .

��Remark 1. It is clear from the proof that to obtain Proposition 2 we do not haveto assume thatf has values inH. It is enough to assume that the values off arebounded inV−1.

5.2. Existence of solutions of HJB equations

Let us begin with the Bellman principle of optimality for Navier-Stokes equa-tions [33,32].

Proposition 3. For every 0 � t � τ � T and x ∈ H,


{∫ τ

t

l (s, X (s; t, x, a(·)) , a(s)) ds + V (τ, X (τ ; t, x, a(·)))}

.

Theorem 4. Assume that Hypothesis 2 is true. Then the value function V is theunique viscosity solution of the HJB equation (5.67)that satisfies (5.68)and (5.69).

Proof. The uniqueness part follows from Theorem 2 and Lemma 1. Here we willonly prove thatV is a viscosity solution and in fact only that the value function is aviscosity supersolution. The subsolution part is very similar and in fact easier. WesetB (x) = B (x, x) throughout this proof. Letψ (t, x) = ϕ (t, x) + δ(t) ||x||20 be atest function. LetV + (ϕ + δ ||·||20) have a local minimum at(t0, x0) ∈ (0, T ) × H.


Step 1. We prove thatx0 ∈ V. For every(t, x) ∈ (0, T ) × H,

V (t, x) − V (t0, x0) � −ϕ (t, x) + ϕ (t0, x0) − δ(t) ||x||20 + δ(t0) ||x0||20 . (5.70)

By the dynamic-programming principle, for everyε > 0 there existsaε (·) ∈ Usuch that, writingXε (s) for X (s; t0, x0, aε(·)) , we have

V (t0, x0) + ε2 >

∫ t0+ε

t0

l (s, Xε (s) , aε(s)) ds + V (t0 + ε, Xε (t0 + ε)) .

Then, by (5.70),

ε2 −∫ t0+ε

t0

l (s, Xε (s) , aε(s)) ds � V (t0 + ε, Xε (t0 + ε)) − V (t0, x0)

� − ϕ (t0 + ε, Xε (t0 + ε)) + ϕ (t0, x0)

− δ(t0 + ε) ||Xε (t0 + ε)||20 + δ(t0) ||x0||20 .

Using the chain rule and the orthogonality relation (2.9) in the above inequality,and then dividing both sides byε, we obtain

ε − 1

ε

∫ t0+ε

t0

l (s, Xε (s) , aε (s)) ds

� − 1

ε

∫ t0+ε

t0

ϕt (s, Xε (s)) ds

+ 1

ε

∫ t0+ε

t0

〈AXε (s) , Dϕ (s, Xε (s))〉 ds

+ 1

ε

∫ t0+ε

t0

〈B (Xε (s)) , Dϕ (s, Xε (s))〉 ds

− 1

ε

∫ t0+ε

t0

〈f (s, aε (s)) , Dϕ (s, Xε (s))〉 ds

+ 1

ε

∫ t0+ε

t0

δ(s) 〈AXε (s) , Xε (s)〉 ds

− 1

ε

∫ t0+ε

t0

δ(s) 〈f (s, aε (s)) , Xε (s)〉 ds

− 1

ε

∫ t0+ε

t0

δ′ (s) ||Xε (s)||20 ds. (5.71)

Now, settingλ = inf t∈[t0,t0+ε0] δ (t) for some fixedε0 > 0, we have forε < ε0

1

ε

∫ t0+ε

t0

δ(s) 〈AXε (s) , Xε (s)〉 ds � λ

ε

∫ t0+ε

t0

||Xε (s)||21 ds.


Therefore it follows thatλ

ε

∫ t0+ε

t0

||Xε (s)||21 ds

� ε − 1

ε

∫ t0+ε

t0


+ 1

ε

∫ t0+ε

t0

ϕt (s, Xε (s)) ds

− 1

ε

∫ t0+ε

t0

〈AXε (s) , Dϕ (s, Xε (s))〉 ds

− 1

ε

∫ t0+ε

t0

〈B (Xε (s)) , Dϕ (s, Xε (s))〉 ds

+ 1

ε

∫ t0+ε

t0

〈f (s, aε (s)) , Dϕ (s, Xε (s))〉 ds

+ 1

ε

∫ t0+ε

t0

δ′ (s) ||Xε (s)||20 ds

+ 1

ε

∫ t0+ε

t0

δ(s) 〈f (s, aε (s)) , Xε (s)〉 ds. (5.72)

The assumptions onl yield∣∣∣∣1ε∫ t0+ε

t0


∣∣∣∣ � C

[1 + 1

ε

∫ t0+ε

t0

‖Xε (s)‖1 ds

]

� C + λ

8· 1

ε

∫ t0+ε

t0

‖Xε (s)‖21 ds. (5.73)

Moreover the assumptions placed onϕ and (2.18) yield

‖A1/2Dϕ (s, Xε (s)) ‖0 � C(‖x0‖0) (5.74)

for ε < ε0 for some fixedε0 > 0. Therefore∣∣〈AXε (s) , Dϕ (s, Xε (s))〉∣∣ =∣∣∣⟨A 1

2 Xε (s) , A12 Dϕ (s, Xε (s))

⟩∣∣∣� ‖A

12 Xε (s) ‖0‖A

12 Dϕ (s, Xε (s)) ‖0

= C(‖x0‖0) ‖Xε (s)‖1

� C + λ

8‖Xε (s)‖2

1 .

Regarding the third term, we have, using (2.12), (2.13) and (5.74),∣∣〈B (Xε (s)) , Dϕ (s, Xε (s))〉∣∣ = |b (Xε (s) , Xε (s) , Dϕ (s, Xε (s)))|� C ||Xε (s)||0 ||Xε (s)||1 ||Dϕ (s, Xε (s))||1� C(‖x0‖0) ‖Xε (s)‖1

� C + λ

8‖Xε (s)‖2

1 .


Finally

| 〈f (s, aε (s)) , Xε (s)〉 |, | 〈f (s, aε (s)) , Dϕ (s, Xε (s))〉 | � C(‖x0‖0). (5.75)

Therefore, using inequalities (5.73)–(5.75) above and the fact thatϕt (s, Xε(s)) andδ′(s) are bounded independently ofε on [t0, t0 + ε0] for someε0 > 0, we obtainfrom (5.72)

λ

ε

∫ t0+ε

t0

||Xε (s)||21 ds � C (5.76)

for some constantC independent ofε. We now takeε = 1n

and setXn (s) =X(s; t0, x0, a1/n(·)). Inequality (5.76) gives then

n

∫ t0+1/n

t0

‖Xn (s)‖21 ds � C

so that, along a sequencetn ∈ (t0, t0 + 1n

),

‖Xn (tn)‖21 � C,

and thus along a subsequence, still denoted bytn, we have

Xn (tn) ⇀ x

weakly in V for somex ∈ V. This also clearly implies weak convergence inH.However, by (2.18),Xn (tn) → x0 strongly (and weakly) inH. Therefore, by theuniqueness of the weak limit inH it follows thatx0 = x ∈ V.

Step 2. The supersolution inequality. We go back to inequality (5.72). By (2.13) (or(2.14)) we have‖Xε(s)‖0 � C(‖x0‖0) = R for everyε ands ∈ [t0, T ]. Thereforeusing (5.62) and (2.20) we obtain

∣∣∣∣1ε∫ t0+ε

t0

[l (s, Xε (s) , aε (s)) ds − l (t0, x0, aε (s))] ds

∣∣∣∣� 1

ε

∫ t0+ε

t0

(σR(|s − t0|) + LR ||Xε (s) − x0||β

)ds � ω1(ε)

for some modulusω1. We will be usingω1 to denote various moduli throughoutthe rest of the proof. Similarly, by the continuity ofϕt , we obtain

∣∣∣∣1ε∫ t0+ε

t0

ϕt (s, Xε (s)) ds − ϕt (t0, x0)

∣∣∣∣� 1

ε

∫ t0+ε

t0

ωt0,x0 (|s − t0| + ||Xε (s) − x0||0) ds � ω1(ε).


Moreover

∣∣∣∣1ε∫ t0+ε

t0

[〈AXε (s) , Dϕ (s, Xε (s))〉 − 〈Ax0, Dϕ (t0, x0)〉]ds

∣∣∣∣� 1

ε

[∫ t0+ε

t0

∣∣〈A (Xε (s) − x0) , Dϕ (s, Xε (s))〉∣∣ ds

+∫ t0+ε

t0

∣∣〈Ax0, Dϕ (s, Xε (s)) − Dϕ (t0, x0)〉∣∣ ds

].

It now follows from (5.74) and (2.20) that

|〈A (Xε (s) − x0) , Dϕ (s, Xε (s))〉|=∣∣∣⟨A 1

2 (Xε (s) − x0) , A12 Dϕ (s, Xε (s))

⟩∣∣∣� ‖Xε (s) − x0‖1 ‖A

12 Dϕ (s, Xε (s)) ‖0

� C(‖x0‖0)‖Xε (s) − x0‖1 � ω1(ε).

By the continuity ofA12 Dϕ we also get

∣∣〈Ax0, Dϕ (s, Xε (s)) − Dϕ (t0, x0)〉∣∣

=∣∣∣⟨A 1

2 x0, A12 [Dϕ (s, Xε (s)) − Dϕ (t0, x0)]

⟩∣∣∣� ‖x0‖1ωϕ,t0,x0 (|s − t0| + ‖Xε (s) − x0‖0) � ω1(ε).

Next we have

∣∣∣∣1ε∫ t0+ε

t0

[〈B (Xε (s)) , Dϕ (s, Xε (s))〉 − 〈B (x0) , Dϕ (t0, x0)〉]ds

∣∣∣∣= 1

ε

∫ t0+ε

t0

|b (Xε (s) , Xε (s) , Dϕ (s, Xε (s))) − b (x0, x0, Dϕ (t0, x0))| ds

� 1

ε

[∫ t0+ε

t0

|b (Xε (s) − x0, Xε (s) , Dϕ (s, Xε (s)))|+ |b (x0, Xε (s) − x0, Dϕ (s, Xε (s)))| ds

+∫ t0+ε

t0

|b (x0, x0, Dϕ (s, Xε (s)) − Dϕ (t0, x0))| ds

].


Now observe that, by (2.11),

|b (Xε (s) − x0, Xε (s) , Dϕ (s, Xε (s)))|� C‖Xε (s) − x0‖1‖Xε (s) ‖1‖Dϕ (s, Xε (s)) ‖1,

|b (x0, Xε (s) − x0, Dϕ (s, Xε (s)))|� C‖Xε (s) − x0‖1‖x0‖1‖Dϕ (s, Xε (s)) ‖1

and, by (2.12)

|b (x0, x0, Dϕ (s, Xε (s)) − Dϕ (t0, x0))|� C‖x0‖0‖x0‖1‖Dϕ (s, Xε (s)) − Dϕ (t0, x0) ‖1

� C‖x0‖0‖x0‖1ωϕ,t0,x0 (|s − t0| + ‖Xε (s) − x0‖0) .

It then follows from (5.74) and (2.20) that∣∣∣∣1ε∫ t0+ε

t0

[〈B (Xε (s)) , Dϕ (s, Xε (s))〉 − 〈B (x0) , Dϕ (t0, x0)〉]ds

∣∣∣∣ � ω1 (ε) .

Also∣∣∣∣1ε∫ t0+ε

t0

[〈f (s, aε (s)) , Dϕ (s, Xε (s))〉 − 〈f (t0, aε (s)) , Dϕ (t0, x0)〉] ds

∣∣∣∣� 1

ε

∫ t0+ε

t0

|〈f (s, aε (s)) , Dϕ (s, Xε (s)) − Dϕ (t0, x0)〉| ds

+ 1

ε

∫ t0+ε

t0

|〈f (s, aε (s)) − f (t0, aε (s)) , Dϕ (t0, x0)〉| ds

� 1

ε

∫ t0+ε

t0

‖f (s, aε (s))‖0 ‖Dϕ (s, Xε (s)) − Dϕ (t0, x0)‖0 ds

+ 1

ε

∫ t0+ε

t0

ωf (|s − t0|) ‖Dϕ (t0, x0)‖0 ds

� ω1 (ε) .

Finally the estimates of the terms containingδ. We have∣∣∣∣2ε∫ t0+ε

t0

[δ(s) 〈AXε(s), Xε(s)〉 − δ(t0) 〈Ax0, x0〉] ds

∣∣∣∣ � ω1(ε)

+2

ε

∫ t0+ε

t0

δ(s)[∣∣∣⟨A 1

2 Xε(s), A12 (Xε(s) − x0)

⟩∣∣∣+∣∣∣⟨A 12 (Xε(s) − x0), A

12 x0

⟩∣∣∣] ds

� ω1(ε) + C

ε

∫ t0+ε

t0

[‖Xε(s)‖1‖Xε(s) − x0‖1 + ‖x0‖1‖Xε(s) − x0‖1] ds

� ω1(ε)


estimating as before. Similarly we obtain∣∣∣∣1ε∫ t0+ε

t0

[δ(s) 〈f (s, aε (s)) , Xε (s)〉 − δ(t0) 〈f (t0, aε (s)) , x0〉]ds

∣∣∣∣ � ω1 (ε)

and ∣∣∣∣1ε∫ t0+ε

t0

δ′ (s) ||Xε (s)||20 ds − δ′ (t0) ||x0||20∣∣∣∣ � ω1 (ε) .

Using all these estimates in (5.72) yields

ω1 (ε) − 1

ε

∫ t0+ε

t0

l (t0, x0, aε (s)) ds

� − ψt(t0, x0) + 〈Ax0, Dψ (t0, x0)〉+ 〈B (x0) , Dϕ (t0, x0)〉 − 1

ε

∫ t0+ε

t0

〈f (t0, aε (s)) , Dψ (t0, x0)〉 ds.

so that

− ψt(t0, x0) + 〈Ax0 + B (x0) , Dψ (t, x0)〉+ 1

ε

∫ t0+ε

t0

[〈f (t0, aε (s)) , −Dψ (t0, x0)〉 + l (t0, x0, aε (s))] ds � ω1 (ε) .

The claim now follows upon taking first the infimum overa ∈ inside the integraland then lettingε → 0. ��

5.3. Applications and possible further developments

The characterization of the value function given byTheorem 4 can be consideredas a starting point for the dynamic-programming analysis of the optimal-controlproblem considered in this paper. It opens up several possibile directions for futureresearch. We outline some of these issues below:

1. One such direction is obtaining verification theorems in the context of viscos-ity solutions. Verification theorems use the HJB equation to give sufficient (and/ornecessary) conditions for optimality. They are natural and easy to prove if the valuefunction is a smooth solution of the HJB equation (see, e.g., [15]). In the nons-mooth case such optimality conditions have been proved in the finite-dimensionalcase (see, e.g., [5], [39, Section 5.3]). For the dynamic programming of Navier-Stokes equation, such a verification theorem is proved in [33,13], [32, Chapter I,Theorem 1.3.12] with a weaker definition of solution.Verification theorems and op-timal synthesis results are also proved for an abstract infinite-dimensional problemin [39, Section 6.5] however these results are rather weak. Moreover the abstractproblem considered there cannot be applied to problems with unbounded non-linearities. Completion of the verification theory for the control of Navier-Stokesequations would require some new ideas, in particular the introduction of appropri-ate generalized sub- and supergradients that would be consistent with our definition


of viscosity solution. Optimality conditions provided by the theory could then beused to construct optimal feedback controls, at least theoretically.

2. Another important direction is the construction ofε-optimal controls andtrajectories, and numerical approximations. There exist several numerical methodsappropriate for solving optimal-control problems for infinite-dimensional systems(in particular for problems governed by nonlinear PDE). One of such promising andrecently popular methods is the POD (proper orthogonal decomposition) method(see for instance [24,29,30]). It is a reduced order method that is based on project-ing the control system onto finite-dimensional subspaces that are spanned by basiselements that contain some characteristics of the expected solution. It also producesfinite-dimensional approximations of optimal controls. We are not aware of any ap-plications of these methods to HJB equations in Hilbert spaces. This field is wideopen. However, some results in this direction seem possible. One is a construction ofan abstract approximation scheme that is connected to discrete dynamic program-ming. The existence and uniqueness results proved in this paper lay a groundworkfor such an attempt.We refer the reader to [5] for an overview of such results in finitedimensions. A different approach to numerical approximation and construction ofε-optimal controls may use the specific nature of our problem, more precisely theweak continuity of the viscosity solution. In the light of this, we can try to approx-imate (uniformly on bounded sets) the unique viscosity solution (and therefore thevalue function) of the infinite-dimensional HJB equation by viscosity solutions ofsuitable finite-dimensional HJB equations connected to finite dimensional controlproblems. Procedures of this type have been used to prove existence of solutions forother infinite-dimensional problems [10, Part IV], [34]. Known numerical methodsfor finite-dimensional HJB equations and dynamic-programming techniques couldthen be used to produceε-optimal controls and trajectories. Of course the case ofNavier-Stokes equations is different from the problems considered in [10,34] butthe idea is worth pursuing.

Sequential quadratic programming is a very promising method where the non-linearity in the Navier-Stokes equation is approximated by Newton’s method toobtain a series of linear problems [20,28]. An approximation of this type appliedto the nonlinearB(x, x) term in the Hamilton-Jacobi equation would lead (for aquadratic cost) to a sequence of approximate problems which fall within the realmof linear-control theory and can be solved effectively by well-established methods.An interesting theoretical question would then be to establish the convergence ofsuch a sequence of approximate solutions to the unique viscosity solution provedin this paper.

3. Optimal control with state constraint is an important component of flowcontrol as it allows us to deal with the task of controlling interesting regions inthe flow domain where we have concentrated turbulence activity, see for example[14]. In the Hamilton-Jacobi framework this will correspond to an initial-boundaryvalue problem (boundary value problem for the stationary case) which would be asubstantial additional step that needs to be addressed in the future.

4.As we pointed out in the introduction, unique solution of the Hamilton-Jacobiequation corresponding to the dynamic programming of the Navier-Stokes equa-tions have been established for a neighborhood of the origin in [4] using different


methods. The uniqueness of viscosity solution proved in this paper may be used toextend this local result to a global one.

5. Finally we would like to mention that we believe that our ideas and meth-ods can be extended to handle the important case of control of three-dimensionalNavier-Stokes equations. The essential reason for the analysis to be limited to twodimensions was the lack of a global-solvability theorem and the related continuous-dependence theorems for the Navier-Stokes equations in three dimensions. Theform of the Hamilton-Jacobi equations however is independent of the spatial di-mensions.

Acknowledgements. A. Swiech was partially supported by NSF grant DMS 0098565.

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Dipartimento di Matematica per le DecisioniEconomiche, Finanziarie e Assicurative

Universita di Roma “La Sapienza”Via del Castro Laurenziano 9, 00161 Roma, Italy

and

Space & Naval Warfare Systems Center, Code D73HSan Diego, CA 92152-5001, USA

and

School of MathematicsGeorgia Institute of Technology

Atlanta, GA 30332, USA

(Accepted January 30, 2002)Published online June 4, 2002 – c© Springer-Verlag (2002)

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