on the exact controllability of hyperbolic magnetic schrödinger equations

Nonlinear Analysis 109 (2014) 319–340

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

On the exact controllability of hyperbolic magneticSchrödinger equationsXiaojun Lu a,b,c,∗, Ziheng Tu d, Xiaofen Lv e

a Department of Mathematics, Southeast University, 211189, Nanjing, Chinab School of Economics and Management, Southeast University, 211189, Nanjing, Chinac BCAM, Alameda de Mazarredo 14, 48009, Bilbao, Bizkaia, Spaind School of Mathematics and Statistics, Zhejiang University of Finance and Economics, 310018, Hangzhou, Chinae Jiangsu Testing Center for Quality of Construction Engineering Co., Ltd, 210028, Nanjing, China

a r t i c l e i n f o

Article history:Received 6 September 2013Accepted 17 June 2014Communicated by Enzo Mitidieri

MSC:35A2335J2535L2035Q4035S1193B0593B07

Keywords:Hamiltonian operatorPseudodifferential operatorsTrace theoremEnergy conservation lawObservability inequalityHilbert Uniqueness MethodUnique continuation theoremCompactness–uniqueness argument

a b s t r a c t

In this paper, we address the exact controllability problem for the hyperbolic magneticSchrödinger equation, which plays an important role in the research of electromagnetics.Typical techniques, such as Hamiltonian induced Hilbert spaces and pseudodifferential op-erators are introduced. By choosing an appropriate multiplier, we proved the observabilityinequality with sharp constants. In particular, a genuine compactness–uniqueness argu-ment is applied to obtain the minimal time. In the final analysis, a suitable boundary con-trol is constructed by the systematic Hilbert Uniqueness Method introduced by J. L. Lions.Compared with the micro-local discussion in Bardos et al. (1992), we do not require thecoefficients belong to C∞. Actually, C1 is already sufficient for the vector potential of thehyperbolic electromagnetic equation.

© 2014 Elsevier Ltd. All rights reserved.

r é s u m é

Dans cet article, on considère le problème de contrôlabilité exacte pour l’équation mag-nétique hyperbolique de Schrödinger, qui joue un rôle important dans la recherche del’électromagnétisme. Les techniques typiques, tels que les espaces de Hilbert induits del’opérateur hamiltonien et des opérateurs pseudo-différentiels, sont introduites. En choi-sissant unmultiplicateur approprié, on a démonstré l’inégalité d’observabilité avec des con-stantes fortes. En particulier, l’argument authentique de compacité-unicité est appliquépour obtenir le temps minimal. Enfin, un contrôle frontière est construit par la méthodesystématique, la méthode hilbertienne de l’unicité introduite par J. L. Lions. Par rapport àla discussion dans Bardos et al. (1992), il n’est pas nécessaire que les coefficients apparti-ennent à C∞. En fait, C1 est déjà suffisante pour le potentiel vecteur de l’équation électro-magnétique hyperbolique.

© 2014 Elsevier Ltd. All rights reserved.

∗ Corresponding author at: Department of Mathematics, Southeast University, 211189, Nanjing, China. Tel.: +86 13813980592.E-mail address: [email protected] (X. Lu).

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320 X. Lu et al. / Nonlinear Analysis 109 (2014) 319–340

1. Introduction to hyperbolic magnetic Schrödinger equations and exact controllability

As is known, amagnetic field is produced by electric fields varying in time, spinning of the elementary particles, ormovingelectric charges, etc. For instance, the earth’s magnetic field is a consequence of the movement of convection currents in theouter ferromagnetic liquid of the core. Nowadays, with the fast development of modern technology, electromagnetic theoryis widely utilized in medical research of organs’ biomagnetism, studying the vortex in the superconductor which carriesquantizedmagnetic flux, and predicting geographical cataclysms, such as earthquakes, volcanic eruptions, and geomagneticreversal.

From the viewpoint ofmathematics, themagnetic field B is a solenoidal vector field whose field line either forms a closedcurve or extends to infinity. In contrast, a field line of the electric field E starts at a positive charge and ends at a negativecharge.

Let A(x) be the vector potential of B, which does not depend on time, that is, B = ∇×A. Evidently,∇ ·B = div rotA = 0.From one of the Maxwell’s equations (µ is the magnetic permeability)

∇ × E = −µ∂B∂t= 0,

we deduce that E = −∇φ, where the scalar φ represents the electric potential. Next we choose an appropriate Lagrangianfor the charged particle in the electromagnetic field (q is the electric charge of the particle, and v is its velocity, m is themass),

L =mv2

2− qφ + qv · A.

In particular, the canonical momentum is specified by the equation

p = ∇vL = mv+ qA.

Then we define the classical Hamiltonian by Legendre transform,

H , p · v− L = mv2 + qA · v−mv2

2− qφ + qv · A

=(p− qA)2

2m+ qφ.

In quantum mechanics, we replace p by−ih∇ , (h is the Planck constant)

H =(ih∇ + qA)2

2m+ qφ.

When we do not consider the influence from the electric field E, then the above Hamiltonian can be simplified as the dif-ferential operator H 2

A , (i∇ + A)2 : H → H∗. H and H∗ will be explained in Section 2. This Hamiltonian operatorphenomenologically describes a number of behaviors discovered in superconductors and quantum electrodynamics (QED).Ginzburg–Landau equations, Schrödinger equations, Dirac equations and the matrix Pauli operator are famous examples inthis respect. For more details, please refer to [1–6].

We are interested in addressing the following variational problem in a suitable function spaceU, whichwill be explainedlater,

minu∈U

12

Ω

|ut |

2− |(i∇ + A(x))u|2 − φ(x)|u|2

dx.

Let the Lagrangian be

L(t, x1, . . . , xn, u, u, ut , ut ,∇u,∇u) , |ut |2− |(i∇ + A(x))u|2 − φ(x)|u|2.

The Euler–Lagrangian equation for L is of the form

∂L

∂u−∂

∂t

∂L∂ut

−

ni=1

∂

∂xi

∂L∂uxi

= 0.

In fact, simple calculation leads to

∂L

∂u= iA · ∇u− A2u− φ(x)u,

∂L

∂ut= ut ,

∂L

∂uxi= −uxi − iaiu.

Consequently, one has

utt +H 2A u+ φ(x)u = 0.

LetΩ ⊂ RN be a bounded open set with a time-independent vector potential A(x). In this work, we mainly address theexact controllability of the hyperbolic magnetic Schrödinger equation in the following form (the modeling process for the

X. Lu et al. / Nonlinear Analysis 109 (2014) 319–340 321

charged particles please refer to [2,7,6,8].)utt +H 2A u = 0 (t, x) ∈ (0, T )×Ω

u = ψ (t, x) ∈ (0, T )× Γu(0, x) = u0, ut(0, x) = u1 x ∈ Ω.

(1)

The concerned function spaces will be explained in the next section. We are interested in the property, i.e. for every initialdata (u0, u1) and every target (uT

0, uT1), whether there exists a boundary control ψ such that the solution u of (1) satisfies

(u(T , x), ut(T , x)) = (uT0, u

T1) for a.e. x ∈ Ω. (2)

Due to the time-reversibility for hyperbolic operators, we can decompose the linear problem (1) into two parts, ua andub. ua is the solution of the homogeneous Dirichlet problemua

tt +H 2A ua= 0 (t, x) ∈ (0, T )×Ω

ua= 0 (t, x) ∈ (0, T )× Γ

ua(T , x) = uT0, u

at (T , x) = uT

1 x ∈ Ω.

Assume that there exists a function ψ such that the solution ub of the problemubtt +H 2

A ub= 0 (t, x) ∈ (0, T )×Ω

ub= ψ (t, x) ∈ (0, T )× Γ

ub(0, x) = u0(x)− ua(0, x), ubt (0, x) = u1(x)− ua

t (0, x) x ∈ Ω

satisfies

ub(T ) = ubt (T ) = 0.

It is evident that u = ua+ ub is the solution of (1) and it satisfies (2). Due to this fact, it is sufficient to consider the null

controllability of (1). Henceforth, we shall assume the target uT0 = uT

1 = 0. Now we give the main theorem.

Theorem 1.1. Assume that A ∈ (C1(Ω))N , and the boundary Γ ∈ C2. When T > 2maxΩ ∥x∥2, then for any initial data(u0, u1) ∈ L2 × H−1, we can find a boundary control ψ ∈ L2([0, T ]; L2(Γ )) such that the hyperbolic magnetic problem (1) isexactly controllable.

In the above theorem, one applies a control on the whole boundary Γ . Next we consider the partial boundary controlproblem. For fixed x0 ∈ RN , let

Γ+ , x ∈ Γ : (x− x0) · ν(x) > 0,Γ− , x ∈ Γ : (x− x0) · ν(x) ≤ 0.

And our control problem is stated in the following form,utt +H 2

A u = 0 (t, x) ∈ (0, T )×Ωu = ψ (t, x) ∈ (0, T )× Γ+u = 0 (t, x) ∈ (0, T )× Γ−u(0, x) = u0, ut(0, x) = u1 x ∈ Ω.

(3)

Apply the same techniques as in Theorem 1.1, and one proves the partial boundary control problem (3).

Theorem 1.2. Assume that A ∈ (C1(Ω))N , and the boundary Γ ∈ C2. When T > 2maxΩ ∥x − x0∥2, then for any initial data(u0, u1) ∈ L2 × H−1, we can find a boundary control ψ ∈ L2([0, T ]; L2(Γ+)) such that the hyperbolic magnetic problem (3) isexactly controllable.

When we include the influence from the electric field E, e.g. replacing H 2A by (i∇ + A(x))2 + φ(x), actually, by applying the

same multipliers and compactness–uniqueness argument, one is able to prove the following fact.

Theorem 1.3. Assume that A ∈ (C1(Ω))N , φ ∈ L∞(Ω) is a nonnegative real function, and the boundary Γ ∈ C2. WhenT > 2maxΩ ∥x − x0∥2, then for any initial data (u0, u1) ∈ L2 × H−1, we can find a boundary control ψ ∈ L2([0, T ]; L2(Γ+))such that the hyperbolic magnetic problem (3) is exactly controllable.

Remark 1.4. Comparedwith the classical multiplierH(x) ·∇v, the newmultiplierH(x) ·HAv has several advantages. On theone hand, it allows to utilize the special quantum structure of the Hamiltonian, such as the magnetic energy conservationlaw, the test matrix for magnetic fieldΞA (which will be explained later), and the Coulomb gauge condition.WhileH(x) ·∇vwill destroy the particular physical structure. On the other hand, it helps to obtain the optimal minimal control time. Incontrast, if we use the multiplier H(x) · ∇v, by the compactness–uniqueness argument, there exist several remainder termswhich can only be estimated by uncertain constants, such as Poincaré constant, which keep us from getting the optimalminimal control time 2∥x − x0∥2. Moreover, this new multiplier can be successfully applied to the discussion of exactcontrollability for magnetic Schrödinger equations.


The rest of the paper is organized as follows. Section 2 is devoted to the description of H 2A -induced function spaces

H10 , H−1 and introduction of H 2

A -pseudodifferential operators in the distributional sense. And the usual compactness–uniqueness argument is applied to demonstrate a generalized Poincaré’s inequality. In Section 3, we use the idea of HUM(Hilbert Uniqueness Method) and apply suitable multipliers to obtain the hidden regularity inequality and observabilityinequality of the adjoint problem of (1), respectively. Readers can also refer to [9–15] for the philosophy of HUM. In orderto prove the minimal time for exact controllability, one uses a genuine compactness–uniqueness argument introduced in[16,17]. In addition, a rigorous proof of the unique continuation theorem for the elliptic operatorL = H 2

A −φ is given in [18].Particularly, for the 1-dimensional case, one can also apply the sidewise energy estimate concerned with the observabilityinequality. In the final analysis, some physical interpretation of the theory and open problems conclude this paper.

2. Prerequisites: basic function spaces and pseudodifferential operators

2.1. H 2A -induced Hilbert spaces

Assume that H(Ω) is a Hilbert space. From the Hamiltonian, we can define the corresponding vector operator

HA , i∇ + A(x) : H(Ω)→ (H(Ω))N ,

where A(x) ∈ RN is the real-valued potential vector. Now we give a function space induced by the vector operator HA.

Definition 2.1. Let A ∈ (L∞(Ω))N , we define a complex function space

H1(Ω) , ω : ω ∈ L2(Ω),HAω ∈ (L2(Ω))N,

which is equipped with the norm

∥ω∥H1 , (∥ω∥2L2 + ∥HAω∥2(L2)N )

12 ,

where

∥(ω1, . . . , ωN)∥(L2)N ,

Nℓ=1

∥ωℓ∥2L2

12

.

Correspondingly, one defines H10 as the closure of D(Ω) in H1, and H−1 as the dual of H1

0 .

Lemma 2.2. Actually, H1 is an equivalent definition of the Sobolev space H1. Consequently, H10 = H1

0 , H−1= H−1 and the

imbeddings H10 → L2 and L2 → H−1 are both dense and compact.

Proof. Indeed, by using the definition of norm in each space, one has

• H1 → H1

∥ω∥2H1 = ∥ω∥2L2 + ∥∇ω∥

2(L2)N

= ∥ω∥2L2 + ∥HAω − Aω∥2(L2)N

≤ ∥ω∥2L2 + 2∥HAω∥2(L2)N + 2∥Aω∥2

(L2)N

≤ (1+ 2N∥A∥2L∞)∥ω∥2L2 + 2∥HAω∥

2(L2)N .

• H1← H1

∥ω∥2H1 = ∥ω∥

2L2 + ∥HAω∥

2(L2)N

= ∥ω∥2L2 + ∥(i∇ + A(x))ω∥2(L2)N

≤ ∥ω∥2L2 + 2∥∇ω∥2(L2)N + 2∥A(x)ω∥2

(L2)N

≤ (1+ 2N∥A∥2L∞)∥ω∥2L2 + 2∥∇ω∥2

(L2)N .

In order to introduce the pseudodifferential operators, we need to make some necessary preparation. First we give aseries of generalized Green’s formulas for the second order operator (iHA)

2 on H2.

Lemma 2.3. For u, v ∈ H2, Γ ∈ C1, one hasΩ

(iHA)2uvdx =

Γ

∂u∂νiHA

· vdΓ −Ω

iHAu · iHAvdx, (4)

X. Lu et al. / Nonlinear Analysis 109 (2014) 319–340 323Ω

(iHA)2uvdx−

Ω

u(iHA)2vdx =Γ

∂u∂νiHA

· vdΓ −Γ

u ·∂v

∂νiHA

dΓ , (5)Ω

(iHA)2udx =

Γ

∂u∂νiHA

dΓ −Ω

A(x) ·HAudx, (6)

where

∂ω

∂νiHA

, (∇ − iA)ω · ν =∂ω

∂ν− iA(x)ω · ν, (7)

and ν is the unit outward normal vector.

Proof. Keep in mind the classical trace theory in [19]. IfΩ is bounded and Γ ∈ C1, then D(Ω) is dense in H2. And the tracemapping v → −→γ v = (γ0v, γ1v) = (v|Γ ,

∂v∂ν

Γ) from H2(Ω) to H

32 (Γ ) × H

12 (Γ ) is linear and continuous. So we prove

these identities on D(Ω). On the one hand,Ω

iHAu · iHAvdx =Ω

∇u · ∇vdx−Ω

iA(x)u · ∇vdx+Ω

i∇u · A(x)vdx+Ω

AATuvdx

=

Ω

∇u · ∇vdx+ ⟨u, iA · ∇v⟩L2 +Γ

iuv · (A · ν)dΓ + ⟨u, i∇ · Av⟩L2 +Ω

AATuvdx.

On the other hand,Ω

(iHA)2uvdx =

Γ

∂u∂ν· vdΓ − 2

Γ

iuv · (A(x) · ν)dΓ

−

Ω

∇u · ∇vdx− ⟨u, iA · ∇v⟩L2 − ⟨u, i∇ · Av⟩L2 −Ω

AATuvdx.

Notice the definition (7), and one concludes the proof of the first identity. The second identity follows immediately whenwe consider the conjugate of the first identity. Finally, the third identity is the special case of v ≡ 1 of the first identity.

Remark 2.4. When A ≡ 0, one has the classical Green’s formulas for Laplacian∆.

Remark 2.5. From the identity (5), we know that H 2A is a self-adjoint differential operator on H2H1

0 . In this case, (7)becomes the usual unit outward normal derivative, i.e.

∂

∂νiHA

=∂

∂ν.

Let ∇j denote ∂∂xj

. Here we introduce an important matrix, i.e. the compatibility matrixΞA,

ΞA ,

ξ11 ξ12, · · · ξ1Nξ21 ξ22 · · · ξ2N...

... · · ·...

ξN1 ξN2 · · · ξNN

(8)

where

ξjk ,

∇j ∇kaj ak

. (9)

Clearly,ΞA is an antisymmetric matrix. In quantum mechanics,ΞA ≡ 0 stands for the case without magnetic field, i.e.

B = rotA = 0.

Once themagnetic field exists, thenΞA = 0. Consequently,ΞA serves as a testmatrix for themagnetic field. In the following,one introduces a considerably significant result, which plays a crucial role in the description of the dual of H1

0 .

Lemma 2.6 (Generalized Poincaré’s Inequality). Let Γ ∈ C1. Then for any ω ∈ H10 , there is a constant C(Ω) > 0 such that

∥ω∥L2 ≤ C(Ω)∥(i∇ + A)ω∥(L2)N . (10)


Proof. (1) A is a constant vector a. In quantummechanics, we say the vector potential A satisfies the famous Coulomb gaugecondition ∇ · A = 0. In this case, one applies the method in Fourier analysis. Assume that Supp ω ⊂ B, B is bounded. For afixed ϵ > 0, divide the frequency into higher and lower parts, then apply Hölder’s inequality, and one has

∥ω∥2L2 =

|a−ξ |≤ϵ

|ω(ξ)|2dξ +|a−ξ |≥ϵ

|ω(ξ)|2dξ

=

|a−ξ |≤ϵ

|ω(ξ)|2dξ +|a−ξ |≥ϵ

|a− ξ |2|ω(ξ)|2

|a− ξ |2dξ

=

|a−ξ |≤ϵ

Bω(x) exp(−ix · ξ)dx

2 dξ + |a−ξ |≥ϵ

|a− ξ |2|ω(ξ)|2

|a− ξ |2dξ

≤ ∥ω∥2L2σ(1)Vol(B)ϵN+ ϵ−2∥(i∇ + a)ω∥2L2 ,

where σ(1) is the volume of a unit ball. Let us choose

ϵ = N

1− β

σ(1)Vol(B), β ∈ (0, 1),

then we have

∥ω∥2L2 ≤1β

σ(1)Vol(B)1− β

2N∥(i∇ + a)ω∥2L2 .

(2) A(x) is not a constant vector. Let us define a semi-norm on H1, i.e.

|ω|H1 , ∥(i∇ + A)ω∥(L2)N = N

j=1

i∂ω∂xj+ aj(x)ω

2L2

12.

First we prove that the above semi-norm is actually a norm on H10 . In fact, let ω be a function from H1

0 (Ω) such that|ω| ˙H1 = 0. Then one has a system of ordinary differential equations inΩ ,

∀j = 1, . . . ,N, i∂ω

∂xj+ aj(x)ω = 0.

Notice the fact ω ∈ H10 , then one has a unique solution ω = 0 inΩ .

Define an equivalent norm in H1, i.e.

∥ω∥H1 , ∥ω∥L2 + |ω|H1 .

We prove the inequality (10) by the method of contradiction. If there does not exist any constant C(Ω) such that, ∀ω ∈ H10 ,

∥ω∥L2 ≤ C(Ω)|ω|H1 ,

then one can find a sequence ωmm from H10 such that

1m∥wm∥L2 > |wm|H1 .

Let

vm ,wm

∥wm∥H1,

then one defines a sequence vmm from H10 such that

∥vm∥H1 = 1, (11)

|vm|H1 <1m. (12)

Since Ω is bounded and open in RN , Γ ∈ C1, then the canonical injection from H1(Ω) to L2(Ω) is compact. According to(11), one can extract a subsequence vµµ from the sequence vmm such that

vµ → v in L2(Ω).


From (12), one has

∀j = 1, . . . ,N, i∂vµ

∂xj+ aj(x)vµ → 0 in L2(Ω).

Since H10 is complete, then one obtains

vµ → v in H10 ,

with

∀j = 1, . . . ,N, i∂v

∂xj+ aj(x)v = 0.

Accordingly, one has

v ≡ 0.

This is impossible since

∥v∥H1 = limµ→∞∥vµ∥H1 = 1.

And we conclude the proof.

2.2. H 2A -induced pseudodifferential operators

According to Lemma 2.6, we introduce an equivalent norm in H10 ,

∥u∥H10

, ∥HAu∥(L2)N , ∀u ∈ H10 . (13)

Lemma 2.3 indicates

(H 2A u, v)L2 = (u, v)H1

0, ∀u ∈ H1

0 such that H 2A u ∈ L2, ∀v ∈ H1

0 . (14)

We know the fact, the imbeddings H10 → L2 and L2 → H−1 are both dense and compact. Consequently, H1

0 → H−1 isdense and compact. As a result, it is reasonable to introduce the duality mapping

H 2A : H

10 → H−1

defined by

⟨H 2A u, v⟩H−1,H1

0, (u, v)H1

0, ∀u, v ∈ H1

0 . (15)

By the Riesz–Fréchet representation theorem, it holds that H 2A is an isometric isomorphism of H1

0 onto H−1. This indicates,D(Ω) is also dense in H−1. Particularly, when A ∈ (C∞(Ω))N , then H 2

A (D(Ω)) = D(Ω). Denoting the compact imbeddingas

I : H10 → H−1.

Then we define a linear and compact mapping

S , (H 2A )−1 I : H1

0 → H10 .

Furthermore, S is positive and self-adjoint. Indeed, for ∀u, v ∈ H10 , according to (14), on the one hand,

(Su, v)H10= ((H 2

A )−1u, v)H1

0= (u, v)L2 .

On the other hand,

(u, Sv)H10= (u, (H 2

A )−1v)H1

0= (u, v)L2 .

Hence,

(Su, v)H10= (u, Sv)H1

0.

Applying the spectral theorem in [20,8] to the compact, self-adjoint and positive linear operator S, we conclude that thespectrum for H 2

A on H10 is discrete, which we denote asΛH 2

A, λkk. And the point spectrum satisfies

0 < λ1 ≤ λ2 ≤ λ3 ≤ · · · → ∞,


with finite multiplicity. In addition, there exists an orthogonal system of complex-valued eigenfunctions φλ(x)λ∈ΛH 2

Ain

H10 , and for each λ ∈ ΛH 2

A,

∥φλ∥L2 = 1.More importantly, φλ(x)λ∈Λ

H 2A

is dense in H10 . Hereafter, we denote by Z the finite combinations of eigenfunctions φλ.

Obviously, Z is dense in H10 .

Remark 2.7. φλ(x)λ∈ΛH 2

Ahas orthogonality in both L2 and H−1. Indeed, for k = l,

0 = (φλk , φλl)H10= ⟨H 2

A φλk , φλl⟩H−1,H10= λk(φλk , φλl)L2 . (16)

While by using the isometric property of H 2A on H1

0 , one has

(φλk , φλl)H−1 = ((H2A )−1φλk , (H

2A )−1φλl)H1

0= λk

−1λl−1(φλk , φλl)H1

0= 0. (17)

Remark 2.8. Moreover, φλ(x)λ∈ΛH 2

Ais also dense in both L2 and H−1 since the density of the imbeddings

H10 → L2 → H−1. (18)

Remark 2.9. LetΩ = (0, π), for the Dirichlet operatori∂

∂x− 1

2

: H10 → H−1,

it is easy to know that 1, 22, 32, . . . ,N2, . . . is the set of eigenvalues which are bounded away from 0. And the associatedorthonormal basis (in the sense of L2-norm) in H1

0 is 2π

sin(x)e−ix,

2π

sin(2x)e−ix,

2π

sin(3x)e−ix, . . . ,

2π

sin(Nx)e−ix, . . ..

Similar as the harmonic operator−∆+ |x|2, if A(x) is not a constant vector, then the eigenfunctions in H10 will not always

take the trigonometric form.

With the above notations, one can define the generalized Fourier transform for f ∈ H−1 as follows:

F f (λ) , ⟨f , φλ⟩H−1,H10. (19)

And the corresponding Fourier series, a unique orthogonal expansion in H−1, is of the form

f (x) =

λ∈ΛH 2

A

F f (λ)φλ(x), (20)

with the RHS (right hand side) converging inH−1. Indeed, for ∀f ∈ H−1, there is a unique uf ∈ H10 such that ⟨f , v⟩H−1,H1

0=

(uf , v)H10for ∀v ∈ H1

0 . Thenλ∈Λ

H 2A

|F f (λ)|2∥φλ∥2H−1 =

λ∈ΛH 2

A

|(uf , φλ)H10|2∥φλ∥

2H−1=

λ∈Λ

H 2A

λ|(uf , φλ)L2 |2∥φλ∥

2L2 <∞.

Remark 2.10. Notice Remarks 2.7 and 2.8, when f ∈ H10 (resp. f ∈ L2), (20) is the unique orthogonal expansion converging

in H10 (resp. in L2). Particularly, when f ∈ H1

0 , then

∥f ∥2H1

0=

λ∈Λ

H 2A

|F f (λ)|2(φλ, φλ)H10=

λ∈Λ

H 2A

λ|F f (λ)|2 <∞,

∥f ∥2L2 =

λ∈ΛH 2

A

|F f (λ)|2(φλ, φλ)L2 =

λ∈ΛH 2

A

|F f (λ)|2 <∞,

∥f ∥2H−1=

λ∈Λ

H 2A

|F f (λ)|2(φλ, φλ)H−1 =

λ∈ΛH 2

A

1λ|F f (λ)|2 <∞.

At the moment, we are ready to introduce the pseudodifferential operators induced by H 2A .


Definition 2.11. Let A ∈ (C∞(Ω))N . Assume that the complex-valued functional F ∈ C(R+) is polynomially bounded. Onedefines a generalized linear pseudodifferential operator as follows:

F

H 2A

: DF

H 2A

⊂ H−1(Ω)→ D ′(Ω), (21)

F

H 2A

u(x) ,

λ∈Λ

H 2A

F(√λ)F u(λ)φλ(x). (22)

The sequence F(√λ)λ∈Λ

H 2Ais referred to as the symbol of F

H 2

A

.

F

H 2A

is defined in the distributional sense. Indeed, for ∀η ∈ D(Ω), since H 2

A (D(Ω)) = D(Ω), then there exists a

unique ηk ∈ D(Ω) such that H 2A · · ·H

2A

k

η = ηk for each k ∈ N. As a result,

(φλ, ηk)L2 = (φλ,H2A · · ·H

2A

k

η)L2

= (H 2A φλ,H

2A · · ·H

2A

k−1

η)L2

= λ(φλ,H2A · · ·H

2A

k−1

η)L2

= λk(φλ, η)L2 .

On the one hand, Hölder’s inequality tells that |(φλ, ηk)L2 | ≤ ∥φλ∥L2∥ηk∥L2 = ∥ηk∥L2 . As a result, |(φλ, η)L2 |λ is a rapidlydecreasing sequence with respect to λ. On the other hand, for ∀u ∈ H−1, there exists a uf ∈ H1

0 such that

F u(λ) = ⟨u, φλ⟩H−1,H10= (uf , φλ)H1

0= ⟨H 2

A φλ, uf ⟩H−1,H10= λ(φλ, uf )L2 .

Apply Hölder’s inequality and one has

|F u(λ)| = |λ(φλ, uf )L2 | ≤ λ∥φλ∥L2∥uf ∥L2 = λ∥uf ∥L2 .

This indicates, |F u(λ)|λ is a polynomially bounded sequence with respect to λ. Since F is also a polynomially boundedfunctional, consequently, the sum on the RHS converges. i.e.

F

H 2A

u(x), η

D′,D=

λ∈Λ

H 2A

F(√λ)F u(λ)(φλ, η)L2 <∞.

Remark 2.12. In particular, when F ∈ C(R+), and F

H 2A

is defined on H1

0 (resp. L2), it is reasonable to substitute D ′(Ω)

by H10 (resp. L2). In this respect, (20) is a natural example. Recall the fact when we consider the unbounded operator H 2

A onH1

0 , then D ′(Ω) is replaced by H−1. For more information about pseudodifferential operators on different manifolds pleaserefer to [21,8].

3. Hilbert uniqueness method

The systematic method, i.e. Hilbert Uniqueness Method, as is indicated by the terminology, is based on Hilbert spacesconstructed by using uniqueness results. In [11,13,14], the authors considered the boundary control and stabilization ofwave equations, Klein–Gordon equations and Petrovsky systems in real Hilbert spaces by applying this method. In [15],J. Vancostenoble and E. Zuazua addressed the exact controllability of the wave and Schrödinger equations perturbed by asingular inverse-square potential. Under suitable geometric conditions, exact boundary controllability is proved in the rangeof subcritical coefficients of the singular potential by the multiplier method.

How to choose suitable multipliers according to the principal elliptic operator is critical. In this section, we are goingto apply the multiplier method to solve the exact controllability of the hyperbolic magnetic equations in complex Hilbertspaces. Mechanically speaking, one is going to exert some external force on the boundary of a bounded magnetic field tomake the object reach its exact target at a given time T . This theory plays a very important part in the research of plasmaphysics and liquid crystal technology.


3.1. Definition of weak solutions

With the terminology introduced in Section 2, we are ready to treat the exact controllability of problem (1) in the timeinterval [L, S], i.e.utt +H 2

A u = 0 (t, x) ∈ (L, S)×Ωu = ψ (t, x) ∈ (L, S)× Γu(L, x) = u(L), ut(L, x) = ut(L) x ∈ Ω.

(23)

As is discussed in Section 1, the time-reversibility property indicates the equivalence between exact controllability and nullcontrollability for (23). In order to define a solution in theweak sense, first we introduce the following homogeneous adjointproblemvtt +H 2

A v = 0 (t, x) ∈ (L, S)×Ωv(t, x) = 0 (t, x) ∈ (L, S)× Γv(L, x) = v(L), vt(L, x) = vt(L) x ∈ Ω.

(24)

Let u, v ∈ C2([L, S];D(Ω)). We multiply (24) with u and integrate by parts.

0 = S

L

Ω

u · (vtt +H 2A v)dxdt

=

Ω

(uvt − utv)dxSL−

S

L

Γ

u ·

∂v

∂νiHA

−∂u

∂νiHA

· vdΓ dt +

S

L

Ω

(utt +H 2A u) · vdxdt

=

Ω

u(S)vt(S)− ut(S)v(S)+ ut(L)v(L)− u(L)vt(L)

dx−

S

L

Γ

ψ ·∂v

∂νiHA

dΓ dt.

Let (u, ut) ∈ C([L, S]; L2 ×H−1) and fix (v(L), vt(L)) ∈ H10 × L2, it is reasonable to define

L SL

v(L), vt(L)

, −ut(L)

u(L)

v(L), vt(L)

+

S

L

Γ

ψ ·∂v

∂νiHA

dΓ dt,

where

−ut(L)

u(L)v(L), vt(L)

,−ut(L), u(L)

,v(L), vt(L)

H−1×L2,H1

0×L2.

Then the identity is rewritten as

L SL

v(L), vt(L)

= −ut(S)

u(S)

v(S), vt(S)

. (25)

Now we give the definition of weak solution of (23).

Definition 3.1. We say that (u, ut) is a weak solution of the non-homogeneous problem (23) if (u, ut) ∈ C([L, S]; L2×H−1)and the identity (25) is satisfied for any S, L ∈ R and every (v(L), vt(L)) ∈ H1

0 × L2 of the homogeneous problem (24).

Clearly, the term SL

Γψ · ∂v

∂νiHAdΓ dt in L S

L must make sense. This indicates, we are looking for an appropriate boundarycontrolψ , which is interactingwith the homogeneous problem in a suitable Sobolev space on the boundarymanifoldM(Γ ).Now we investigate the homogeneous problem (24) to gain more insightful information of the outward normal derivativewith respect to iHA.

3.2. Representation of solutions and energy conservation laws for the adjoint problem

In thewhole spaceR, it is easy to check that, ifω(t, x) = F(x+t)+G(x−t) solves the classicalwave equationωtt−ωxx = 0,then v(t, x) = exp(ix)F(x + t) + exp(−ix)G(x − t) is the solution of vtt + (i ∂∂x − 1)2v = 0. One can discover that, thepropagation speed is 1. We are interested to know the structure of the solution for (24).

Theorem 3.2. For any given initial data (v(L), vt(L)) ∈ H10 × L2 and S ∈ R, the homogeneous problem (24) has a unique

solution such that

v ∈ C([L, S];H10 )

C1([L, S]; L2)

C2([L, S];H−1).


Proof. At first, we assume sufficient regularity for the solution. Apply the generalized Fourier transforms (19) and (20) tothe homogeneous equation (24), then one has

λ∈ΛH 2

A

(F v(λ))tt + λF v(λ)

φλ(x) = 0.

Since φλ(x)λ∈ΛH 2

Ais a complete orthogonal system of complex-valued eigenfunctions in H−1, then for ∀λ ∈ ΛH 2

A,

(F v(λ))tt + λF v(λ) = 0.

As a result,

F v(λ) = cos(t√λ)F v(L)(λ)+

sin(t√λ)

√λ

F vt(L)(λ).

Since (v(L), vt(L)) ∈ H10 × L2, then in H1

0 ,

v(x) =

λ∈ΛH 2

A

F v(λ)φλ(x).

Consequently, Definition 2.11 gives the unique converging form by pseudodifferential operators,

v(t, x) = cost

H 2A

v(L)+

sint

H 2A

H 2A

vt(L). (26)

Notice Remark 2.12, then it can be checked that v ∈ C([L, S];H10 ). Other regularity follows immediately from (26) through

symbol calculus. Indeed, if the initial data are composed of finite combinations of eigenfunctions, then elliptic regularitytheory gives u ∈ C∞(R;H2(Ω)).

Now we turn to the energy conservation property for (24).

Definition 3.3. We define the energy of homogeneous problem (24) as

E(v)(t) , ∥vt∥2L2 + ∥v∥

2H1

0.

Lemma 3.4. The energy conservation law holds for the homogeneous problem (24).

Proof. Here we apply the symbol calculus for pseudodifferential operators. All the steps are rigorously deduced by thedefinition of (21) and (22) induced by H 2

A on H10 . From (26), one has

∥vt(t, ·)∥2L2 =−H 2

A sint

H 2A

v(L)+ cos

t

H 2A

vt(L)

2L2

=

H 2A sin

t

H 2A

v(L)

2L2+

costH 2A

vt(L)

2L2

− 2Re

H 2A sin

t

H 2A

v(L), cos

t

H 2A

vt(L)

L2,

∥v(t, ·)∥2H1

0=

H 2A cos

t

H 2A

v(L)

2L2+

sintH 2A

vt(L)

2L2,

+ 2Re

H 2A sin

t

H 2A

v(L), cos

t

H 2A

vt(L)

L2.

As a result,

E(v)(t) =H 2

A sint

H 2A

v(L)

2L2+

H 2A cos

t

H 2A

v(L)

2L2

+

sintH 2A

vt(L)

2L2+

costH 2A

vt(L)

2L2

= (H 2A v(L), v(L))L2 + ∥vt(L)∥

2L2 = ∥v(L)∥

2H1

0+ ∥vt(L)∥2L2

= E(v)(L).


3.3. Hidden regularity inequality

Recall the definition of L SL . In order to know more about the boundary control ψ , one needs to investigate the term ∂v

∂ν

carefully. By the density method, traditional trace theory tells us that ∂v∂ν(t) ∈ H

12 (Γ ). In the following, we will give an

important inequality which indicates the continuity of ∂v∂ν(t)with respect to initial data.

Theorem 3.5 (Hidden Regularity Inequality). For any given L, S ∈ R and initial data (v(L), vt(L)) ∈ H10 × L2, the outward

normal derivative defined in the form of (7) satisfies

∂v

∂ν∈ L2(L, S; L2(Γ ))

and S

L

Γ

∂v∂ν2 dΓ dt ≤ C(1+ (S − L))E(v)(L).

In particular, for the 1-dimensional case, let Ω = (−1, 1), then the above inequality can be rewritten as S

L(|vx(t,−1)|2 + |vx(t, 1)|2)dt ≤ C(1+ (S − L))E(v)(L).

3.3.1. Deduction of the first identityIn order to prove the above theorem, first we introduce an important identity. In the following, we consider the problem

(24) with initial data from Z × Z due to its density in H10 × L2. To begin with, we apply the multiplier H(x) · HAv to (24),

where the vector field H ∈ C1(Ω;RN). One decomposes the following integral into two parts, i.e. S

L(vtt +H 2

A v,H(x) ·HAv)L2dt = S

L(vtt ,H(x) ·HAv)L2dt

(I)

+

S

L(H 2

A v,H(x) ·HAv)L2dt (II)

.

Apply the generalized Green’s formula in Lemma 2.3, then one has

(I) = (vt ,H(x) ·HAv)L2

SL−

S

L(vt ,H(x) ·HAvt)L2dt

= (vt ,H(x) ·HAv)L2

SL−

S

L

Ω

vt ·H(x) ·HAvt

dxdt.

(II) = S

L

Ω

H 2A v ·

H(x) ·HAv

dxdt

=

S

L

Γ

−∂v

∂ν·

H(x) ·HAv

dΓ dt +

S

L

Ω

HAv ·HA

H(x) ·HAv

dxdt

(III)

.

And we calculate the term (III).

(III) = S

L

Ω

j

i∇jv + aj(x)v

·

i∇j + aj(x)

k

hk(x) ·i∇kv + ak(x)v

dxdt

=

S

L

Ω

j

i∇jv + aj(x)v

· i∇j

k


dxdt

+

S

L

Ω

j

i∇jv + aj(x)v

· aj(x)

k


dxdt

=

S

L

Ω

j,k

i∇jv + aj(x)v

· i∇jhk(x) ·

i∇kv + ak(x)v

dxdt

+

S

L

Ω

j,k

i∇jv + aj(x)v

· hk(x) ·

i∇j + aj(x)

i∇kv + ak(x)v

dxdt.


Combining (I), (II) and (III), one has the identity, S

L

Γ

∂v

∂ν·

H(x) ·HAv

dΓ dt

= (vt ,H(x) ·HAv)L2

SL−

S

L

Ω

vt ·H(x) ·HAvt

dxdt

+

S

L

Ω

j,k

i∇jv + aj(x)v

· i∇jhk(x) ·

i∇kv + ak(x)v

dxdt

+

S

L

Ω

j,k

i∇jv + aj(x)v

· hk(x) ·

i∇j + aj(x)

i∇kv + ak(x)v

dxdt. (27)

Moreover, for the 1-dimensional case, identity (27) is simplified as

− i S

L|vx|

2h(x)dt1−1= 2i Im

S

L

1

−1vtt ·

h(x) ·HAv

dxdt +

S

L

1

−1HAv · i∇h(x) ·HAvdxdt. (28)

3.3.2. Proof of Theorem 3.5 in 1-dimensional caseChoose g(x) ∈ C2([−1, 1]) such that ∇g = 0 and define h(x) , ∇g

|∇g| . It is evident that h(x) ∈ C1([−1, 1]) and h = ν onΓ . From (28), one has,

− i S

L|vx|

2h(x)dt1−1= 2i Im(vt , h(x) ·HAv)L2

SL− 2i Im

S

L

1

−1vt ·

h(x) ·HAvt

dxdt

− i S

L

1

−1∇h(x)|HAv|

2dxdt. (29)

On the one hand,

−i S

L|vx|

2h(x)dt1−1= −i

S

L(|vx(t,−1)|2 + |vx(t, 1)|2)dt.

On the other hand, S

L

1

−1vt ·

h(x) ·HAvt

dxdt =

S

Lvt · h(x) · ivtdt

1−1−

S

L

1

−1∇h(x) · vt · ivtdxdt

−

S

L

1

−1∇vt ·

h(x) · ivt

dxdt +

S

L

1

−1vt · a(x) ·

h(x) · vt

dxdt

= i S

L

1

−1∇h(x)|vt |2dxdt +

S

L

1

−1HAvt · h(x) · vtdxdt.

Transfer the term SL

1−1 HAvt · h(x) · vtdxdt on the RHS to the LHS, then we have

2i Im S

L

1

−1vt ·

h(x) ·HAvt

dxdt = i

S

L

1

−1∇h(x)|vt |2dxdt.

And (29) can be rewritten as

12

S

L(|vx(t,−1)|2 + |vx(t, 1)|2)dt = −Im(vt , h(x) ·HAv)L2

SL+

12

S

L

1

−1∇h(x)(|vt |2 + |HAv|

2)dxdt. (30)

Consider the first term on the RHS,Im(vt , h(x) ·HAv)L2

SL

≤ 2 supx|h(x)|E(v)(L).

As for the second term on the RHS, it is clear that12

S

L

1

−1∇h(x)(|vt |2+|HAv |

2)dxdt ≤ sup

x|∇h(x)|(S − L)E(v)(L).

In the final analysis, S

L(|vx(t,−1)|2 + |vx(t, 1)|2)dt ≤ C(H)(1+ (S − L))E(v)(L).

This concludes the proof for the 1-dimensional case.


3.3.3. Deduction of the second identityFor the multidimensional case, (27) is far from enough. It is necessary to develop a new identity to obtain more

information. First we decompose (27) into four parts, i.e. S

L

Γ

∂v

∂ν·

H(x) ·HAv

dΓ dt = (vt ,H(x) ·HAv)L2

SL

(I)

−

T

0

Ω

j

vt · hj(x) ·i∇jvt + aj(x)vt

dxdt

(II)

+

S

L

Ω

j,k

i∇jv + aj(x)v

· i∇jhk(x) ·

i∇kv + ak(x)v

dxdt

(III)

+

S

L

Ω

j,k

i∇jv + aj(x)v

· hk(x) ·

i∇j + aj(x)

i∇kv + ak(x)v

dxdt

(IV)

.

We treat the second term (II).

(II) = S

L

Ω

vt ·H(x) ·HAvt

dxdt

=

j

i S

L

Ω

∇jvt · hj(x) · vtdxdt +

j

S

L

Ω

vt · hj(x) · aj(x) · vtdxdt +

j

i S

L

Ω

∇jhj(x) · |vt |2dxdt

=

S

L

Ω

vt ·H(x) ·HAvt

dxdt + i

S

L

Ω

∇ · H(x)

· |vt |

2dxdt.

This leads to the following fact,

2 Im S

L

Ω

vt ·H(x) ·HAvt

dxdt =

S

L

Ω

∇ · H(x)

· |vt |

2dxdt.

Next we turn to the third term (III). Let

ΘH ,

∇1h1 ∇1h2, · · · ∇1hN∇2h1 ∇2h2, · · · ∇2hN...

... · · ·...

∇Nh1 ∇Nh2, · · · ∇NhN

.Then we represent (III) in the sense of matrix calculus, i.e.

(III) = −i S

L

Ω

HAv ×ΘH ×H TA vdxdt,

where the quadratic form is

HAv ×ΘH ×H TA v = (i∇1v + a1v, . . .)

∇1h1 ∇1h2, · · · ∇1hN∇2h1 ∇2h2, · · · ∇2hN...

... · · ·...

∇Nh1 ∇Nh2, · · · ∇NhN

i∇1v + a1vi∇2v + a2v

...

i∇Nv + aNv

.Then we consider the fourth item (IV), which is the most complicated part. (In the following, [, ] is the Lie bracket.)

(IV) = S

L

Ω

j,k

i∇jv + aj(x)v

· hk(x) ·

i∇j + aj(x)

i∇kv + ak(x)v

dxdt

=

S

L

Ω

j,k

i∇jv + aj(x)v

· hk(x) · i∇k

i∇jv + aj(x)v

dxdt

+

S

L

Ω

j,k

i∇jv + aj(x)v

· hk(x) · ak(x) ·

i∇jv + aj(x)v

dxdt

(V)

+

S

L

Ω

j,k

i∇jv + aj(x)v

· hk(x) ·

i∇j + aj(x), i∇k + ak(x)

vdxdt.


It is worth noticing that the item (V) is purely real. When we consider only the imaginary part of (IV), (V) disappears andone has

Im(IV) = −12

S

L

Γ

H(x) · ν

·

HAv

2dΓ dt +12

S

L

Ω

∇ · H(x)

HAv

2dxdt+ Im

S

L

Ω

ivHAv × ΞA × HTdxdt,

whereΞA is the test matrix of magnetic field defined in (8), i.e.

HAv × ΞA × HT= (i∇1v + a1v, . . . , i∇Nv + aNv)

ξ11 ξ12, · · · ξ1Nξ21 ξ22 · · · ξ2N...

... · · ·...

ξN1 ξN2 · · · ξNN

h1h2...hN

.Combining (I), (II), (III) and (IV), one has a new identity,

−12

S

L

Γ

∂v∂ν

2 · H(x) · νdΓ dt

= Im(vt ,H(x) ·HAv)L2

SL−

12

S

L

Ω

∇ · H(x)

·

|vt |

2−

HAv

2dxdt− Re

S

L

Ω

HAv ×ΘH ×H TA vdxdt − Re

S

L

Ω

vHAv × ΞA × HTdxdt. (31)

3.3.4. Proof of Theorem 3.5 in multidimensional caseSinceΩ is compact, then it can be covered with a finite number of neighborhoods Ok ⊂ RN , k = 1, 2, . . . ,m, in which

there is a unique O1 satisfying O1 ∩ Γ = ∅. Similar as the procedure in the 1-dimensional case, we choose ζ1 = 0 in O1and ζk ∈ C1(Ok) such that ζk = ν on Ok ∩ Γ for k = 2, . . . ,m. Let θk ∈ C2

0 (Ok), k = 1, . . . ,m be a partition of unity,corresponding to the covering Okk. Then we define

H(x) ,

k

θkζk

Ω.

It is easy to check that H ∈ C1(Ω) and H = ν on Γ . Keep in mind the energy conservation law, now we begin to estimateeach term on the RHS of (31).(1) For the first term,Im(vt ,H(x) ·HAv)L2

SL

≤ Ω

vt(S) ·H(x) ·HAv(S)

dx+

Ω

vt(L) ·H(x) ·HAv(L)

dx

≤12(1+max

x∥H∥2)

Ω

|vt(S)|2dx+Ω

|HAv(S)|2dx

+12(1+max

x∥H∥2)

Ω

|vt(L)|2dx+Ω

|HAv(L)|2dx

= 2(1+maxx∥H∥2)E(v)(L).

(2) For the second term,12

S

L

Ω

∇ · H(x)

·

|vt |

2−

HAv

2dxdt ≤ maxx|∇ · H|(S − L)E(v)(L).

(3) For the third term, by taking into account the inequality for Frobenius norm in matrix analysis, i.e.

∥ΘH ×H TA v∥2 ≤ ∥ΘH∥F∥HAv∥2,

where

∥ΘH∥F ,

j,k

|∇jhk|2 1

2,

one hasRe S

L

Ω

HAv ×ΘH ×H TA vdxdt

≤ S

L

Ω

∥HAv∥2∥ΘH∥F∥HAv∥2dxdt.

≤ maxx∥ΘH∥F

S

L

Ω

|HAv|2dxdt

≤ 2maxx∥ΘH∥F (S − L)E(v)(L).


(4) For the last term, recall the generalized Poincaré’s inequality in Lemma 2.6, and one hasRe S

L

Ω

vHAv × ΞA × HTdxdt ≤ S

L

Ω

|v|∥HAv∥2∥ΞA∥F∥H∥2dxdt

≤12max

x(∥ΞA∥F∥H∥2)

S

L

Ω

(|HAv|2+ |v|2)dxdt

≤ 2Cp maxx(∥ΞA∥F∥H∥2)(S − L)E(v)(L). (32)

Consequently, L

S

Γ

∂v∂ν

2 dΓ dt ≤ C(A,H)(1+ (S − L))E(v)(L).

Up to now, Theorem 3.5 is completely proved. Compared with the inequality for the 1-dimensional case, here the constantdepends on both H and the magnetic potential A in the multidimensional case.

3.4. Well-posedness of the hyperbolic magnetic problem

Now we turn to our previous question. Recall the term SL

Γψ · ∂v

∂νdΓ dt in L S

L . Theorem 3.5 shows that for any giveninitial data (v(L), vt(L)) ∈ H1

0 × L2, then ∂v∂ν∈ L2loc(R; L

2(Γ )). If ψ ∈ L2loc(R; L2(Γ )), then L S

L is well defined. We are eagerto know, whether problem (23) is well-posed with such a control ψ . Indeed, one has the following positive answer.

Theorem 3.6. Given any (u(L), ut(L)) ∈ L2 × H−1 and any ψ ∈ L2([L, S]; L2(Γ )), problem (23) has a unique solutionsatisfying (u(S), ut(S)) ∈ L2×H−1 and the linear mapping (u(L), ut(L), ψ) −→ (u(S), ut(S)) is continuous from L2×H−1×L2([L, S]; L2(Γ )) into L2 ×H−1 with respect to these topologies.

Proof. Recall the identity (25). i.e.

L SL

v(L), vt(L)

= −ut(S)

u(S)

v(S), vt(S)

.

FromTheorem3.5, it follows that for any L, S ∈ R, the linear operatorL SL is bounded onH1

0×L2. Pay attention to Theorem3.2

and the energy conservation law in Lemma 3.4, due to time-reversibility, then one finds the linear mapping

(v(S), vt(S)) −→ (v(L), vt(L))

is an isometric isomorphism of H10 × L2 onto itself. Consequently, the linear form

(v(S), vt(S)) −→ L SL (v(L), vt(L))

is also bounded onH10×L

2. According to the Riesz–Fréchet representation theorem, there exists a unique pair (ut(S), u(S)) ∈H−1 × L2 satisfying (25). The existence and uniqueness are proved.

Nextwe show that ∥(ut(S), u(S))∥H−1×L2 is uniformly bounded in each finite time interval I . Indeed, by applying Hölder’sinequality, Theorem 3.5 and the energy conservation law in Lemma 3.4, one has− ut(S)

u(S)

v(S), vt(S)

≤ − ut(L)

u(L)v(L), vt(L)

+ S

L

Γ

ψ ·∂v

∂νdΓ dt

≤ ∥(−ut(L), u(L))∥H−1×L2∥(v(L), vt(L))∥H1

0×L2 + ∥ψ∥L2(I,L2(Γ ))

∂v∂νL2(I,L2(Γ ))

≤

∥(−ut(L), u(L))∥H−1×L2 + C(I)∥ψ∥L2(I,L2(Γ ))

∥(v(L), vt(L))∥H1

0×L2

=

∥(−ut(L), u(L))∥H−1×L2 + C(I)∥ψ∥L2(I,L2(Γ ))

∥(v(S), vt(S))∥H1

0×L2 .

The definition of norm for the tensor product operator indicates,

∥(ut(S), u(S))∥H−1×L2 ≤ ∥(−ut(L), u(L))∥H−1×L2 + C(I)∥ψ∥L2(I,L2(Γ )).

This demonstrates the continuous dependence of the solution on (u(L), ut(L), ψ).

3.5. Observability inequality

At themoment, we begin to search an appropriateψ ∈ L2loc(R; L2(Γ )) such that (23) is exactly controllable. First of all, we

need to construct a Hilbert space with a suitable norm concerned with the outward normal derivative of the homogeneousadjoint system (24). Wewill prove that, under certain conditions, ∥ ∂v

∂ν∥L2loc (R;L

2(Γ )) is actually an equivalent norm onH10 ×L2.


Theorem 3.7. Let T ∗ , 2maxΩ ∥x∥2. For any given L, S ∈ R and initial data (v(L), vt(L)) ∈ H10 × L2, then when S − L > T ∗,

the outward normal derivative defined by (7) satisfies

((S − L)− T ∗)E(v)(L) ≤ C(Ω) S

L

Γ

∂v∂ν2 dΓ dt.

In particular, in the 1-dimensional case, let Ω = (−1, 1), the above inequality can be rewritten as

((S − L)− T ∗)E(v)(L) ≤ C(Ω) S

L(|vx(t,−1)|2 + |vx(t, 1)|2)dt.

3.5.1. Proof of Theorem 3.7 in 1-dimensional case: multiplier methodIt is sufficient to prove the estimate for (u(L), ut(L)) ∈ Z×Z , the general case then follows by a density argument. Similar

as the proof of Theorem 3.5, different identities will be applied accordingly. Moreover, a sidewise energy estimate, which isintroduced by E. Fernández-Cara and E. Zuazua in [9] can also be applied. Interesting readers can check this fact.

Pay attention to the identity (28) for the 1-dimensional case. Let h(x) , x, then (28) is simplified as

− i S

L(|vx(t,−1)|2 + |vx(t, 1)|2)dt

= 2i Im(vt , x ·HAv)L2

SL− 2i Im

S

L

1

−1vt ·

x ·HAvt

dxdt − i

S

L

1

−1|HAv|

2dxdt. (33)

For the second term on the RHS of (33), integrating by parts, one has S

L

1

−1vt ·

x ·HAvt

dxdt = i

S

L

1

−1|vt |

2dxdt + S

L

1

−1HAvt · x · vtdxdt.

Consequently,

2i Im S

L

1

−1vt ·

x ·HAvt

dxdt = i

S

L

1

−1|vt |

2dxdt.

According to energy conservation law in Lemma 3.4, the identity (33) can be rewritten as

12

S

L(|vx(t,−1)|2 + |vx(t, 1)|2)dt = −Im(vt , x ·HAv)L2

SL+ (S − L)E(v)(L). (34)

Now we investigate the first term on the RHS of (34). Applying Hölder’s inequality, one hasIm(vt , x ·HAv)L2

SL

≤ Im 1

−1vt(S) · x ·HAv(S)dx

+ Im 1

−1vt(L) · x ·HAv(L)dx

≤ E(v)(S)+ E(v)(L) ≤ 2E(v)(L).

Consequently, when S − L > 2, we have the observability inequality,

((S − L)− 2)E(v)(L) ≤12

S

L(|vx(t,−1)|2 + |vx(t, 1)|2)dt.

3.5.2. Step 1 of Proof of Theorem 3.7 in multidimensional case: multiplier methodLet H(x) , x, thenΘH is the identity matrix and the identity (31) is simplified as

−12

S

L

Γ

∂v∂ν

2 · x · νdΓ dt = Im(vt , x ·HAv)L2

SL−

N2

S

L

Ω

|vt |

2−

HAv

2dxdt−

S

L

Ω

|HAv|2dxdt − Re

S

L

Ω

vHAv × ΞA × HTdxdt. (35)

Next we apply the multiplier v to (24). By decomposing the following integral into two parts and applying the generalizedGreen’s formula (4) in Lemma 2.3, we have S

L(vtt +H 2

A v, iv)L2dt = −iΩ

vt · vdxSL+ i

S

L

Ω

|vt |2dxdt − i

S

L

Ω

|HAv|2dxdt. (36)


Combining (35) and (36), we obtain a new identity,

12

S

L

Γ

∂v

∂νiHA

2 · x · νdΓ dt = − Im(vt , x ·HAv)L2

SL

(I)

− Imvt ,

N − 12

iv

L2

SL

(II)

+(S − L)E(v)(L)

+ Re S

L

Ω

vHAv × ΞA × HTdxdt (III)

. (37)

Actually, we can have a more refined estimate. SinceΩ

x ·HAv · vdx = −iN∥v∥2L2 +Ω

x ·HAv · vdx,

then vt , x ·HAv +

N − 12

iv

L2≤

12maxΩ∥x∥2∥vt∥2L2 +

12max

Ω∥x∥2

x ·HAv +N − 1

2iv2L2

=12maxΩ∥x∥2∥vt∥2L2 +

12max

Ω∥x∥2

∥x ·HAv∥

2L2 +

N − 12

v

2L2−(N − 1)N

2∥v∥2L2

≤

12maxΩ∥x∥2∥vt∥2L2 +

12max

Ω∥x∥2

(maxΩ∥x∥2)2∥HAv∥

2L2

≤ max

Ω∥x∥2E(v)(L).

As a result,

|(I)+ (II)| ≤ 2maxΩ∥x∥2E(v)(L).

As to (III), by applying Schwartz’s inequality, we haveRe S

L

Ω

vHAv × ΞA × HTdxdt ≤ S

L

Ω

|v|∥HAv∥2∥ΞA∥F∥H∥2dxdt

≤ ϵmaxx(∥ΞA∥F∥H∥2)

S

L

Ω

|HAv|2dxdt

+14ϵ

maxx(∥ΞA∥F∥H∥2)

S

L

Ω

|v|2dxdt

≤ ϵmaxx(∥ΞA∥F∥H∥2)(S − L)E(v)(L)

+14ϵ

maxx(∥ΞA∥F∥H∥2)

S

L

Ω

|v|2dxdt. (38)

Let η , maxx(∥ΞA∥F∥H∥2), consequently,

(1− ϵη)(S − L)− 2max

Ω∥x∥2

E(v)(L) ≤

maxΩ∥x∥2

2

S

L

Γ

∂v∂ν

2 dΓ dt +η

4ϵ

S

L

Ω

|v|2dxdt. (39)

Remark 3.8. In quantummechanics, ∥ΞA∥F stands for the intensity of themagnetic field∇×A. For the 1-dimensional case,∥ΞA∥F ≡ 0, so there is no influence from the magnetic field. In fact, many typical vector potentials also satisfy ∥ΞA∥F ≡ 0,such as A = η(|x|2)x, where η ∈ C∞(R3).

3.5.3. Step 2 of Proof of Theorem 3.7 in multidimensional case: exact controllability in a minimal timeIn this section, one applies the compactness–uniqueness argument introduced in [16,10,17].With thismethod, the lower

order terms can be absorbed by the boundary integral.

Lemma 3.9. There exists a positive constant C(ϵ,Ω) such that S

L

Ω

|v|2dxdt ≤ C(ϵ,Ω) S

L

Γ

∂v∂ν

2dΓ dt. (40)


Proof. We prove it by the method of contradiction. Assume that (40) does not hold, then there exists a sequence of initialdata (vm0 , v

m1 ) ∈ H1

0 × L2 such that the corresponding sequence of solutions vmm of the homogeneous adjoint problem(24) satisfies S

L

Ω

|vm|2dxdt = 1 for allm; (41) S

L

Γ

∂vm∂ν

2dΓ dt ≤1m. (42)

Recall the norm in H1([L, S] ×Ω), e.g.

∥ω∥H1([L,S]×Ω) , ∥ω∥L2t (L2)+ ∥ωt∥L2t (L

2) + ∥HAω∥L2t (L2).

Since ω ∈ C([L, S];H10 ), according to Lemma 2.6, one can define an equivalent norm in H1((L, S)×Ω) as

∥ω∥H1([L,S]×Ω) , ∥ωt∥L2t (L2) + ∥HAω∥L2t (L2)

.

According to (39) (41) and (42), for (1−ϵη)S−L > 2maxΩ ∥x∥2, one knows that (vm0 , vm1 ) is bounded inH1

0 ×L2, therefore,

(vm0 , vm1 ) (v∗0 , v

∗

1) in H10 × L2. (43)

Since the canonical injection from H1([L, S] × Ω) to L2([L, S] × Ω) is compact, then one can extract a subsequence, (forconvenience’s sake, we still use the notation vmm), such that

vm v∗ in H1([L, S] ×Ω), (44)

vm → v∗ in L2([L, S] ×Ω), (45)

where v∗ is the solution corresponding to the limit initial data (v∗0 , v∗

1). (For rigorous proof please see [22,23].) Furthermore,for any φ ∈ H1([L, S] ×Ω)with φ(L, x) = φ(S, x) = 0, we have

0 = S

L

Ω

((vmtt − v∗

tt)φ +H 2A (v

m− v∗)φ)dxdt

= −

S

L

Ω

(vmt − v∗

t )φtdxdt − S

L

Γ

∂vm

∂ν−∂v∗

∂ν

φdΓ dt +

S

L

Ω

(HAvm−HAv

∗)HAφdxdt. (46)

From (44), we have

limm

S

L

Γ

∂vm

∂ν−∂v∗

∂ν

φdΓ dt = 0. (47)

This gives

∂vm

∂ν

∂v∗

∂νin L2([L, S] × Γ ). (48)

According to the lower semicontinuity property for weak convergence, from (42) we deduce that

∂v∗

∂ν= 0 in L2([L, S] × Γ ). (49)

Definition 3.10. A weak solution v ∈ L2([L, S];H10 ) with vt ∈ L2([L, S]; L2) and vtt ∈ L2([L, S];H−1) of vtt +H 2

A v = 0 is

called invisible if it satisfies v|Γ = ∂v∂ν

Γ= 0. The set of all invisible solutions is denoted as N .

Remark 3.11. N is a finite dimensional subspace of L2([L, S]×Ω). Indeed, when (1−ϵη)S−L > 2maxΩ ∥x∥2, for ∀v ∈ N ,it holds, S

L

Ω

|HAv|

2+ |vt |

2dxdt ≤ C(η, ϵ, L, S)

S

L

Ω

|v|2dxdt. (50)

Since the canonical injection fromH1([L, S]×Ω) to L2([L, S]×Ω) is compact, according to the Riesz Theorem, (50) indicatesthat dim(N ) is finite. Since 1

S − Lexpit

√λ φλ,

1

S − Lexp−it

√λ φλ

λ∈Λ

H 2A


forms a basis satisfying (24) in L2([L, S] ×Ω), so N has a basis of the form 1S − L

expit√λi φλi ,

1

S − Lexp−it

√λi φλi

ni=1.

Consequently, due to the elliptic regularity theory, for ∀v ∈ N , v ∈ C∞([L, S];H2).

In the following, we show thatN is indeed 0. First, we introduce an important unique continuation theorem concernedwith the magnetic operator H 2

A . The specific proof based on the multiplier method is given in [18].

Theorem A (Unique Continuation Theorem). For N ≥ 2, let ω ∈ H2(B1) be a solution of the elliptic problem

H 2A ω = φ(x)ω in B1,

where B1 is a unit ball and the complex function φ ∈ L∞(RN). If ω vanishes in a neighborhood of x0 ∈ B1, then ω ≡ 0 in B1.

From the above theorem, we can deduce the following result.

Corollary 3.12. Let Ω be a bounded open domain in RN with the boundary Γ ∈ C2. Let ω ∈ H2 be a solution of

H 2A ω = φω inΩ

ω =∂ω

∂iHA

= 0 on Γ .

Then ω ≡ 0 inΩ .

Proof. Let B be an arbitrarily small open ball such that

Γ ∩ B = ∅.

Set

Ω1 , Ω ∪ B,

and define

ω1 ,

ω inΩ;0 in B \Ω.

It is sufficient to verify that ω1∈ H2. Denote by ω1

j , ω1jk the extension by zero to Ω1 of the derivatives ∇jω, ∇j∇kω,

j, k = 1, . . . ,N . Then ωj, ωjk ∈ L2(Ω1) and it is necessary to demonstrate that, for ∀ζ ∈ D(Ω1),Ω1ω1∇jζdx = −

Ω1ω1

j ζdx,

and Ω1ω1

j ∇kζdx = −Ω1ω1

jkζdx.

Indeed, since ω1j = ω

1jk ≡ 0 outside ofΩ , ζ ≡ 0 on Γ \ (Γ ∩ B) and ω = ∂ω

∂iHA≡ 0 on Γ ∩ B, one has

Ω1ω1∇jζdx =

Ω

ω∇jζdx =Γ

ωζνjdΓ −Ω

(∇jω)ζdx

=

Γ∩B

ωζνjdΓ −Ω

(∇jω)ζdx = −Ω

(∇jω)ζdx = −Ω1ω1

j ζdx,

and Ω1ω1

j ∇kζdx =Ω

∇jω∇kζdx =Γ

∇jωζνkdΓ −Ω

(∇k∇jω)ζdx

=

Γ∩B∇jωζνkdΓ −

Ω

(∇k∇jω)ζdx = −Ω

(∇k∇jω)ζdx = −Ω1ω1

jkζdx.


If N = 0, at least there should exist a nonzero φλi such that

H 2A φλi = λiφλi , φλi |Γ =

∂φλi

∂ν

Γ= 0.

The above unique continuation property implies that φλi ≡ 0. Consequently, N ≡ 0. This is in direct contradiction to thefact that

SL

Ω|v∗|2dxdt = 1.

Theorems 3.5 and 3.7 indicate, for T > 2maxΩ ∥x∥2, T

0

Γ

∂v∂ν2 dΓ dt

defines an equivalent norm in H10 × L2 with respect to Definition 2.1. While Theorems 3.2 and 3.5 demonstrate that, for any

given T ∈ R and initial data (v(0), vt(0)) ∈ H10 × L2, there exists a unique solution

v ∈ C([0, T ];H10 )

C1([0, T ]; L2)

C2([0, T ];H−1)

for the homogeneous problem (24) and the outward normal derivative defined by (7) satisfies

∂v

∂ν∈ L2([0, T ]; L2(Γ )),

which is continuous with respect to the initial data. If we choose

ψ ,∂v

∂ν∈ L2([0, T ]; L2(Γ ))

and consider the problem (23) with the initial data (u(T ), ut(T )) = 0, then Theorem 3.6 shows that problem (23) has aunique solution satisfying

(u0, u1) , (u(0), ut(0)) ∈ L2 ×H−1.

And the linear mapping

(u(T ), ut(T ), ψ) −→ (u(0), ut(0))

is continuous from L2 ×H−1 × L2([L, S]; L2(Γ )) into L2 ×H−1 with respect to these topologies. Let (v0, v1) be the initialdata of (24). Thus, in a unique fashion, one can define a linear and bounded mapping

S : H10 × L2(Ω) −→ H−1 × L2(Ω),(v0, v1) → (u1,−u0).

(51)

It is evident that, if S is surjective, then ψ , ∂v∂ν

is an appropriate boundary control which drives (u0, u1) ∈ L2 × H−1 torest. In the following lemma we will show this fact.

Lemma 3.13. Let T > 2maxΩ ∥x∥2, then S is an isomorphism from H10 × L2 onto H−1 × L2.

Proof. Recall the Lions–Lax–Milgram lemma in [23].

Lemma 3.14. Let V be a real or complex Hilbert space, and V ∗ be its dual. V is continuously imbedded into V ∗. If there exists aκ > 0 such that A ∈ L (V , V ∗) satisfies the so-called V -elliptic condition, i.e.

⟨Aω,ω⟩V ∗,V ≥ κ∥ω∥2V , ∀ω ∈ V ,

then A is an isomorphism from V onto V ∗.

It is sufficient to check the V -elliptic condition for S . Taking into account of the data u(T ) = ut(T ) = 0, we multiplythemagnetic equation (23) by the solution v of the adjoint homogeneous problem (24). By applying the generalized Green’sformula, we have

0 = T

0

Ω

v · (utt +H 2A u)dxdt

=

Ω

(vut − vtu)dxT0−

T

0

Γ

v ·

∂u∂νiHA

−∂v

∂νiHA

· udΓ dt +

T

0

Ω

(vtt +H 2A v) · udxdt

=

Ω

(v(T )ut(T )− vt(T )u(T )+ v1u(0)− v0ut(0))dx+ T

0

Γ

∂v∂ν2 dΓ dt.


Hence,

⟨S (v0, v1), (v0, v1)⟩H−1×L2,H10×L

2 =

T

0

Γ

∂v∂ν2 dΓ dt.

For T > 2maxΩ ∥x∥2, the observability result in Theorem 3.7 gives the V -elliptic condition. Consequently, S is anisomorphism from H1

0 × L2 onto H−1 × L2.

Remark 3.15. Theorems 1.2 and 1.3 can be proved similarly by using the same multipliers as in Theorem 1.1. Here we donot go into details of the proofs. Actually, by applying the transmutation method introduced in [24,25], one can check thenull controllability results for magnetic heat equations and magnetic Schrödinger equations.

Remark 3.16. During the proof, we find that, in a magnetic field with large ∥ΞA∥F , it is difficult to impose some exteriorforce on the boundary to influence the interior activity. Astronomically speaking, when the solar wind with coronal massejections encounters Earth’s magnetosphere, most of the radioactive particles are deflected around the earth instead ofimpacting the atmosphere or the earth’s surface, although some leakage occurs, resulting in auroras and Van Allen belts.

Remark 3.17. It is really challenging to investigate the control theory in the field of quantummechanics. Many interestingproblems, such as the exact controllability of Maxwell’s equations, and nonlinear Ginzburg–Landau equations, are to beaddressed. More references are to be found in [7,26].

Acknowledgments

This project was suggested by Prof. Enrique Zuazua during the first author’s post-doctoral research in BCAM from2010–2011. This project is supported byNatural Science Foundation of Jiangsu Province BK20130598, Fundamental ResearchFunds for the Central Universities in Southeast University (No. 3207012208 (2012) and No. 3207013210 (2013)). Thisproject is also supported by Grant MTM2008-03541 and MTM2011-29306 of the MICINN (Spain), the ERC Advanced GrantFP7-246775 NUMERIWAVES, the ESF Research Networking Programme OPTPDE and the Grant PI2010-04 of the BasqueGovernment. The authors thank BCAM and Key Laboratory of Systems and Control (Chinese Academy of Sciences) forproviding the nice research environment. Moreover, the authors also express their deep gratitude to the referees for theircareful reading and valuable suggestions.

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on the exact controllability of hyperbolic magnetic schrödinger equations

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