Controllability of Schrödinger equation with a nonlocal term

DOI:10.1051/cocv/2013052 www.esaim-cocv.org

CONTROLLABILITY OF SCHR ¨ODINGER EQUATION WITH A NONLOCAL TERM

Mariano De Leo

, Constanza S´ anchez Fern´ andez de la Vega

and Diego Rial

Abstract. This paper is concerned with the internal distributed control problem for the 1D Schr¨odinger equation,i ut(x, t) =−uxx+α(x)u+m(u)u,that arises in quantum semiconductor models.

Herem(u) is a non local Hartree–type nonlinearity stemming from the coupling with the 1D Poisson equation, andα(x) is a regular function with linear growth at infinity, including constant electric fields.

By means of both the Hilbert Uniqueness Method and the contraction mapping theorem it is shown that for initial and target states belonging to a suitable small neighborhood of the origin, and for distributed controls supported outside of a fixed compact interval, the model equation is controllable.

Moreover, it is shown that, for distributed controls with compact support, the exact controllability problem is not possible.

Mathematics Subject Classification. 93B05, 81Q93, 35Q55.

Received March 19, 2012. Revised December 18, 2012.

Published online August 29, 2013.

1. Introduction

We are mainly concerned with the internal distributed controllability for the following 1D Schr¨odinger equation

iu_t=−u_xx+α(x)u+m(u)u, x∈R, t >0, (1.1)

u(x,0) =u₀(x), (1.2)

posed in the Sobolev space H= {φ ∈ H¹(R) :

μ(x)|φ|² <∞}, where μ is a positive regular function that coincides with|x|away from the origin. Here, the non linearitym(u) is of non local nature:

m(φ)(x) =

(x, y)|φ(y)|²dy, (1.3)

Keywords and phrases.Nonlinear Schr¨odinger–Poisson, Hartree potential, constant electric field, internal controllability.

1 Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Guti´errez 1150 (1613) Los Polvorines, Buenos Aires, Argentina.mdeleo@ungs.edu.ar

2 IMAS – CONICET and Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabell´on I (1428) Buenos Aires, Argentina.csfvega@dm.uba.ar; drial@dm.uba.ar

Article published by EDP Sciences c EDP Sciences, SMAI 2013

ET AL.

where the kernel satisfies the estimate |(x, y)| ≤ μ(y). This choice is motivated for the self-consistent 1D Schr¨odinger–Poisson equation used in quantum semiconductor theory

iu_t=−uxx+u

|x| ∗

D − |u|²

(1.4) whereD ∈C^∞(R) denotes the fixed positively charged background orimpurities, see [8] and references therein for semiconductor models. After a suitable rearrangement of terms the Hartree potential reads

|x| ∗

D − |u|²

=

(|x−y| −μ(x))(D(y)− |u(y, t)|²)dy+μ(x)

(D(y)− |u(y, t)|²)dy

=

(|x−y| −μ(x))D(y)dy−

(|x−y| −μ(x))|u(y, t)|²dy+

μ(x)

D_L¹_(R)− u(·, t)²_L2(R)

.

Introducing a := D_L¹_(R)− u(·, t)²_L2(R), F(x) :=

(|x−y| −μ(x))D(y)dy, and (x, y) = μ(x)− |x−y|, equation (1.4) thus becomes

iu_t(x, t) =−uxx(x, t) + (aμ(x) +F(x))u(x, t) +m(u(x, t))u(x, t), takingα(x) :=aμ(x) +F(x) we show that the evolution equation (1.4) becomes (1.1).

We note that in the 1D case the kernel μ(x) is not bounded nor integrable so the classic theory developed in [1] does not apply and we refer to [3] for details on the well posedness. In this article we will consider a slightly extended version in which the term aμ(x) is replaced by a regular function α(x) ∈ C^∞(R), with at most linear growth at infinity (i.e.with the asymptoticsα(x)∼C^±xforx∼ ±∞), in order to include constant electric fieldsα(x) =qx. We note that due to the regularity requirements of the unique continuation technique displayed in Lemma3.2, the regular functionα(x) appears as a regularized approximation of a locally constant electric field, which is modelled with a continuous piecewise linear function. It is also worth to mention that since the impurities give rise to a bounded potential

F(x) =

(|x−y| − |x|)D(y)dy,

and hence enters in the model equation as a bounded multiplication operator, and since our results are still valid for bounded perturbations, there is no loss of generality in restricting ourselves to the caseF ≡0.Let us finally mention that results on controllability with local nonlinearities as|u|^2σuare widely developed, see [5,11], and therefore local nonlinearities will not be taken into consideration.

The problem of exact internal controllability of equation (1.1)–(1.2) is usually described as the question of finding a control functionh∈L²(0, T,H) and its associated state functionu∈C(0, T,H) such that

iu_t=−u_xx+α(x)u+m(u)u+ψ(x)h(x, t), x∈R, t∈(0, T), (1.5)

u(x, t₀) =u₀(x), u(x, T) =u_T(x) (1.6)

where T > 0 is a given target time and u₀ and u_T are the given initial and target states respectively, and ψ:R→Ris a givenC¹function that localizes the control to Supp(ψ). The problem of distributed controllability for Schr¨odinger equations of nonlinear type appears often in nonlinear optics, see for instance [4,9]. There are several results on controllability of the Schr¨odinger equation, for a review on this topic we refer to [13].

In this paper we discuss the internal distributed controllability for the problem iu_t=−u_xx+α(x)u+m(u)u, x∈R, t >0, u(x, t₀) =u₀(x)

and present results concerning two different situations depending on the support of the control: on one hand controls that are supported outside a compact interval, in which case we shall give positive results, and on the other hand localized controls, in which case we shall give a non controllability result.

We start dealing with a distributed control given byψ(x)h(x, t) where ψ∈C¹(R) satisfies:

ψ(x) =

1 for|x| ≥R+ 1

0 for|x| ≤R. (1.7)

We thus show that for a given 0 < T there exist a (small) constant δ such that for every u₀, u_T ∈ H with u₀H,u_TH< δ there exists a controlh(x, t)∈L²(0, T,H) such that the nonlinear problem (1.5)–(1.6) has a unique solutionu∈C(0, T,H).

We then turn to the case in which ψ ∈ C¹ is compactly supported and show that for both α= μ (linear operator with a discrete spectrum) and α(x) = x(constant electric field, which has a continuous spectrum), the linear system is not exactly controllable. More precisely we show that for any fixed finite time T >0 and any fixed target stateu_T ∈ Hthere exist an open bounded intervalΩ and an initial stateu₀ ∈ H, such that for anyψ with Supp(ψ)⊂Ω, there is no control function ψ(x)h(x, y), with h∈L²(0, T,H), and no constant C=C(T, Ω) such that

iu_t=−u_xx+α(x)u+ψ(x)h, x∈R, t∈[0, T], u(x,0) =u₀(x), u(x, T) =u_T(x)

withh_L¹_(0,T,L²_(Ω))≤C(T, Ω) (u0_H+u_TH).

The paper is organized as follows. We set the problem in Section 1. In Section 2, we deal with the existence of dynamics and establish useful estimates for the related evolution. Section 3 is devoted to the problem in which the control vanishes inside an open bounded interval. We start studying the linear system for which we prove global controllability in the spaceH; we then prove the local controllability for the nonlinear system (1.5). In Section 4, we deal with the non controllability result for compactly supported controls.

2. Preliminaries

In this section we shall collect some results concerning spectral properties for the operator−∂_x²+α(x).Since most of the estimates refer to different functional spaces we list them below:

• H¹(R) :={φ∈L²(R) :φ_x∈L²(R)}.

• L²_μ(R) := {φ :μ^1/2φ∈ L²(R)} where μ is a regular even function satisfying 1≤μ(x), and μ(x)≡ |x| for

|x| ≥2.

• H:=H¹(R)∩L²_μ(R) withφ²_H=φ_x²_L2+φ²_L2 µ. 2.1. Existence of dynamics

To start with we consider the auxiliar operatorL₊ defined by L₊:H → H

φ→L₊(φ) :=

−∂_x²+|x|

φ. (2.1)

Although this operator does not enter directly in our model, because of the loss of regularity of|x| in the origin, it provides the workspaceHand also it possesses useful spectral properties, easily deduced from the ones of the Airy function, that are needed for the proof of the non controllability result of Theorem4.1.

Lemma 2.1. The operator L₊ satisfies the following properties:

(a) It is self-adjoint inL²(R).

ET AL.

(b) It has a discrete spectrum 0<λ₁< . . . <λ_N +∞.

(c) It has a countable set of orthonormal (with respect to L²) eigenfunctions {φ_N :N∈N} ⊆ H satisfying λ^−1/4_N

Ω|(φ_N)_x|^{2 1/2}≤C(Ω), (2.2)

where Ωis an arbitrary bounded interval.

Remark 2.2. Self-adjointness ofL₊ and the existence of both a discrete spectrum,{0<λ₁ <λ₂ < . . .},and an orthonormal basis of eigenfunctions,{φ_N}_N∈N⊆ H,follows directly from [2] where by means of variational methods it is only shown thatL⁻¹₊ is a compact operator. However, the non-controllability result relies on some special feature of the eigenfunctions, given by claim (c), that are not considered there and we shall give an alternative proof.

Proof. We first notice that the related quadratic form verifiesφ;L₊φ=φx²_L2+|x|^1/2φ²_L2 and this is an equivalent norm forH,from where we recover the self-adjointness ofL₊. The operatorL₊has an explicit spectral decomposition expressed in terms of the Airy function Ai, defined as the solution of −Ai_xx(x) +xAi(x) = 0 such that Ai(+∞) = 0, as follows. Let 0 < z₀ < z₁ < . . . +∞, and 0 < w₀ < w₁ < . . . +∞ be the zeros of Ai(−x) and Ai(−x) respectively, and takeλ_2N =z_N,λ_2N+1 =w_N,and φ_2N(x) =c_2NAi(|x| −λ_2N), φ_2N+1(x) =c_2N+1sgn(x)Ai(|x|−λ_2N+1),wherec_N is a (bounded) sequence of normalization constants. A direct computation shows thatL₊(φ_N) =λ_Nφ_N.This gives the spectral decomposition ofL₊.Since for|x| ∼+∞it happens that|x|−λ_N >0,each eigenfunctionφ_N inherits the decaying properties of the Airy function near +∞

where it behaves as e^−r3/2.

In order to get claim (c) we take profit of the integral expression for the Airy function and its derivative, withx=−|x|,

Ai(x) = (2π)^−1/2|x|^1/2

e^{i|x|3/2(k3/3−k)}dk Ai(x) = (2π)^−1/2

ike^{i|x|3/2(k3/3−k)}dk

from where, by means of the stationary phase method, we deduce the asymptotics

|Ai(x)| ≤C(M)|x|^1/4 (2.3)

valid forx≤ −M, and also an estimate for the eigenvalues

λ_N ∼N^2/3. (2.4)

LetM be such thatΩ⊆[−M, M],from estimate (2.4) there existsN₀such that, forN > N₀,λ_2N−λ_N > M.

Then, forx∈Ωone has|x| −λ_2N < M−λ_2N <−λ_N.Using (2.3) we conclude|φ_2N(x)|<λ^1/4_N ,and therefore

(φ_N)_xL²_(Ω)≤(2M)^1/2λ^1/4_N/2. This finishes the proof.

As a direct consequence of previous result we get the existence of dynamics for the operatorL_μ defined by L_μ:H → H

φ→L_μ(φ) :=

−∂_x²+μ(x)

φ (2.5)

whereμ(x) is a regular even function satisfyingμ(x)≡ |x|for|x| ≥2 and max{1,|x|} ≤μ(x)≤1 +|x|.

Lemma 2.3. The operator L_μ is well defined and verifies (a) It is self adjoint.

(b) It has a discrete spectrum 0< λ₁ ≤λ₂≤λ_N +∞, and a countable set of orthonormal (with respect to L²) eigenfunctions{ϕ_N :N ∈N} ⊆ H.

Proof. It follows directly from the inequalities

φ;L₊φ ≤ φ;L_μφ ≤ φ²_L2+φ;L₊φ, and the compact embeddingH→L².

Notice that previous estimates yield the asymptotics

λ_N ∼N^2/3.

In order to develop the observability inequality we need to build some appropriate Sobolev spaces, related to the operatorL_μ defined by (2.5). This is done as follows. Let{ϕ_N}_N∈N be the orthonormal basis of L² given by Lemma2.3and, forφ∈L²,letφbe the Fourier coefficient:φ(N ) :=φ;ϕ_N.We then set fork= 0,1,2 the Hilbert spacesW^k:={φ∈L²:

N≥0λ^k_Nφ(N) ²<∞},with the inner product ψ;φ_W^k :=

N≥0

λ^k_Nψ(N )^∗φ(N ). (2.6)

LetF ⊂W⁰be the set of finite linear combinations of{ϕN}N∈N. Then fork=−3,−2,−1 the inner product (2.6) is well defined. We then defineW^k as the Hilbert space obtained from the closure ofF with the norm induced by·;·_W^k. We have thatL_μ:W^k →W^k−2 is an isometry:L_μw_W^k−2 =w_W^k. Being L_μ positive, we have L^1/2_μ :W^k→W^k−1 which is also an isometry:L^1/2_μ w_W^k−1=w_W^k.

We finally mention thatW⁰=L², W¹ =H, W² =D(L_μ), the domain of the operatorL_μ :W²→L², and W⁻¹=H,with compact embeddings

W²⊂W¹⊂W⁰⊂W⁻¹⊂W⁻². (2.7)

Remark 2.4. Since for anyψ∈W^k andφ∈W^−k we have ψ;φL² =

N≥0

ψ(N) ^∗φ(N )

=

N≥0

λ^k/2_N ψ(N )^∗λ^−k/2_N φ(N )

≤ ψ_Wkφ_W−k, we also have fork=−2,−1,0,1,2 that

W^k

=W^−k.

We now turn to the general situationL:=−∂_x²+α(x),where α(x)∈C^∞(R) is a regular function verifying α_x, α_xx∈L^∞,and also the asymptotics

x→±∞lim α(x)

μ(x) =C^±. (2.8)

The following lemma states precisely the self-adjointness result.

Lemma 2.5. Let α∈ C^∞(R) satisfying (2.8). Then L: H → H defined by L :=−∂_x²+α(x) is self-adjoint, and therefore−iLgenerates a strongly continuous group of unitary operators inL²(R).

ET AL.

Proof. To this purpose we first show thatL is a closed operator. Let ϕ ∈C₀^∞(R) and (φ_n;L(φ_n))∈ H × H a sequence such that (φ_n;L(φ_n)) → (φ;ψ) in H × H, since ϕ;φ_xx −(φ_n)_xx = ϕ_xx;φ−φ_n → 0 and ϕ;α(φ−φ_n)=sgn(α)|α|^1/2ϕ;|α|^1/2(φ−φ_n) →0 we thus have ϕ;L(φ−φ_n) → 0, and consequently we concludeϕ;ψ−Lφ=ϕ;ψ−Lφ_n+ϕ;L(φ_n−φ) →0.This shows thatL:H → H is a closed operator.

Since L_μ := −∂_x²+μ(x), with μ(x) ≥ 1 we deduce that L_μ ≥ I (the identity operator). For ϕ, ψ ∈ H we introduce the (well defined) bilinear form Q(φ, ψ) := φx;ψ_x+φ;α(x)ψ. We now establish two useful estimates

|Q(φ;ψ)| ≤ |φ_x;ψ_x|+|φ;αψ|

≤(1 +αμ⁻¹L^∞)|φ;L_μψ|

≤(1 +αμ⁻¹L^∞)L^1/2_μ φL²L^1/2_μ ψL²

|Q(Lμφ;ψ)− Q(φ;L_μψ)|=|φ; [Lμ:L]ψ|

≤ |φ; (μ−α)_xxψ|+ 2|(μ−α)_xφ;ψ_x|

≤((μ−α)_xxL^∞+ 2(μ−α)_xL^∞)L^1/2_μ φ_L²L^1/2_μ ψ_L² where we have used the identityL^1/2μ ϕ²_L2 =ϕ_x²_L2+ϕ²_L2

µ.Applying Theorem X.36’ in [10] we obtain that Lis a essentially self-adjoint operator inH,since it is closed, it follows thatLis self adjoint.

2.2. Scattering properties for constant electric fields

The non controllability result, see Theorem4.4, for a constant electric fieldL_e:=−∂_x²−x,follows from a well- knownL¹−L^∞estimate for the groupU_e(t) generated by−iL_e,which depends upon a result of Avron–Herbst, see [12] for details.

Lemma 2.6. The operator L_e is essentially self-adjoint onS(R)and

U_e(t) = e^−it3e^itxe^{−i(p2t+t2p)} (2.9) wherep=−i∂_xis the momentum operator.

Remark 2.7. Identity (2.9) says that except for phase factorsU_e(t)φ(x) is obtained by first translating byt² units to the right and then applying the free particle group e^it∂x2

Corollary 2.8. Forφ∈L¹(R)we have the following estimate:

U_e(t)φ_L^∞ ≤C|t|^−1/2φ_L¹. 2.3. Estimates for the evolution

Lemma 2.5guarantees that−iLgenerates a groupU(t). In the sequel we will exhibit useful bounds for the evolution related to both the homogeneous and inhomogeneous problem.

Lemma 2.9. Let U(t)be the group generated by−iL whereL:=−∂_x²+αin H. Then

• (U(t)φ)_xL²≤ φx_L²+|t|αxL^∞φ_L².

• U(t)φL²_µ≤ φL²_µ+ 2^1/2|t|^1/2φ^1/2_L2 φ_x^1/2_L2 +|t|α_xL^∞φL².

• U(t)φH≤ φH

1 +|t| · μx−α_xL^∞ .

Proof. Letu(t) =U(t)φ,sinceuverifiesiu_t=−u_xx+αuandu²_H=u²_L2

µ+u_x²_L2 we have d

dt

u_x;u_x

L² = 2Re u_xt;u_x

L²

= 2Re

u_x;−iα_xu

L²

d dt

u;μu

L² = 2Re u_t;μu

L²

= 2Re

u_x;iμ_xu

L²

d dt

u;u

H= 2Re

u_x;i(μ_x−α_x)u

L².

The inequalities are obtained by means of a standard ODE argument given by the following lemma. Details are

given due to the lack of a suitable reference.

Lemma 2.10. Let y: [0, T]→[0,+∞)be an L¹ function satisfying the inequalityy²(t)≤y²(0) +C_t

0y(s)ds for some constant C >0.Then y(t)≤y(0) +Ct/2.

Proof. Let w(t) := _t

0y(s)ds and z(t) :=

y²(0) +Cw(t). Then ˙z(t) ≤ C/2 and therefore y(t) ≤ z(t) ≤

z(0) +Ct/2.

We now turn our attention to the non linear term in equation (1.5), and give the following estimates.

Lemma 2.11. Let m:H →L^∞(R)be given by m(φ)(x) =

(x, y)|φ(y)|²dy

where|(x, y)| ≤μ(y)and|_x(x, y)| ≤C. Then forφ, φ₁∈ Hthe following estimates hold.

• m(φ)_L^∞ ≤ φ²_L2

• m(φ)φ−m(φ₁)φ₁µH≤3/2

φ²_H+φHφ1H+φ1²_H

φ−φ₁H. Proof. It is a straightforward computation and will be omitted.

We now turn to the non homogeneous problem (1.5) and give similar estimates in the lemma below, which in turn express the global well posedness of the problem.

Lemma 2.12. Let T > 0 be fixed, and let u ∈ C(0, T,H)∩C¹(0, T,H) be a solution of (1.5) with fixed h∈L²(0, T,H)andψ∈C¹(R)such that ψ,and ψ_x∈L^∞(R). Then we have the following estimates:

• u_L^∞_(0,T,L²_(R))≤ u0_L²_(R)+T^1/2ψ_L^∞h_L²_(0,T,H).

• u_xL^∞_(0,T,L²_(R))≤ u0_H¹+h_L²_(0,T,H)T^3/2C(u₀, ψ).

• u_L^∞_(0,T,L²_µ(R)) ≤ u0_L²_µ+h_L²_(0,T,H)T^3/2C(u₀, ψ).

• u_L^∞_(0,T,H)≤ u0_H+h_L²_(0,T,H)T^3/2C(u₀, ψ).

Proof. Using estimates for the linear and nonlinear term, given in Lemmas2.9and2.11respectively, the proof relies on a procedure similar to the one displayed in Lemma 2.9and will be omitted.

ET AL.

3. Controllability

3.1. Linear system

We start this section taking into consideration the controllability of the linear problem, which throughout this section means the existence of a controlh(x, t) such that the unique solution of the related non homogeneous linear equation

iu_t(x, t) =Lu(x, t) +ψ(x)h(x, t) (3.1)

u(x,0) =u₀(x) x∈R (3.2)

satisfies u(x, T) = u_T(x), for given T > 0 and u₀, u_T(x) ∈ H, where L := −∂²_x+α(x) is the operator of Lemma 2.5, andψ is defined in (1.7). The main result is given in the following theorem; its proof is based on the Hilbert Uniqueness Method (HUM), requires some technicalities, which we shall first develop, and will be delayed until the end of this subsection.

Theorem 3.1 (Global controllability: linear case). Let T > 0 be given. Then there exists a bounded linear operator G:H × H → L²(0, T,H)such that for any u₀, u_T ∈ H the system (3.1)–(3.2), with h=G(u₀, u_T), admits a solutionu∈C(0, T,H)satisfying u(x, T) =u_T.

As we stated before, we need first to present the ingredients to apply the HUM. To do this, we consider the corresponding adjoint problem inH:

iv_t(x, t) =Lv, (3.3)

v(x,0) =v₀(x). (3.4)

Let Λ :H → H denote the usual isomorphism between the real spacesH and H defined by Λ(v) = v,·H. Given v₀ ∈ H, let v be the solution of equation (3.3). Then, take h(·, t) = Λ⁻¹(ψv(·, t)) and consider the

problem

iw_t(x, t) =Lw+ψ(x)h(x, t),

w(x, T) =u₁(x), (3.5)

which we split into the two problems:

iw⁽¹⁾_t (x, t) =Lw⁽¹⁾,

w⁽¹⁾(x, T) =u₁(x), (3.6)

and

iw_t⁽²⁾(x, t) =Lw⁽²⁾+ψ(x)h(x, t),

w⁽²⁾(x, T) = 0. (3.7)

Clearly, w =w⁽¹⁾+w⁽²⁾. As usual with the HUM procedure, given v₀ ∈ H the initial condition of equation (3.3), we define the linear operatorS:H→ Hby

S(v₀) =−iw⁽²⁾(·,0) (3.8)

wherew⁽²⁾ is the solution of (3.7).

If we can show that S is an isomorphism, then the inverse image by S of −iu0+iw⁽¹⁾(·,0), is the initial condition for equation (3.3) that will provide the sought controlh=Λ⁻¹(ψv(·, t)).

This is shown by establishing the observability inequality of system (3.3) in H which we describe in the following lemma.

Lemma 3.2. Let ψbe aC¹ function defined by (1.7). There exists a constantC >0such that for allv₀∈ H, the solutionv of (3.3)–(3.4)satisfies

_T

0

ψv(., t)²_Hdt≥Cv₀²_H. (3.9)

The proof of the observability inequality (3.9) is quite similar to the one given by L. Rosier and B. Zhang in [11]. We repeat most of the construction given in that paper for the sake of completeness.

In order to prove Lemma3.2we begin by proving the corresponding observability inequality inH. We recall the isomorphismL_μ:H → H,L_μ=−∂_x²+μ. Consider the Schr¨odinger equation

iw_t(x, t) =Lw+P(w), (3.10)

w(x,0) =w₀(x), (3.11)

whereP(w) =L⁻¹_μ [ν, L_μ](w), withν :=α−μ.

Lemma 3.3. Following assertions are true:

(a) ∂:W^k →W^k−1 is a bounded operator fork= 0,1.

(b) Forg∈L^∞ such thatg∈L^∞the related multiplication operatorg:W^k→W^k is bounded fork= 0,1,−1.

(c) P:W^k→W^k is a bounded operator for k= 0,1,−1.

(d) w_L^∞_(0,T,W^k)≤C(T)w_0W^k for k= 0,1,−1.

Proof. For k = 0 claims (a) and (b) are evident. Claim (a) for k = 1 is obtained from k = 0 by duality W⁻¹= (W¹) (see Rem.2.4). Claim (b) fork= 1 follows from the estimate:

gφ_H≤ g_L^∞φ_L²+g_L^∞φ_L²+g_L^∞φ_L²_µ.

By duality we also get claim (b) fork =−1. Claim (c) is a consequence of claims (a) and (b) applied to the identity

L⁻¹_μ [ν, L_μ] =L⁻¹_μ (2ν_x∂_x+ν_xx) where we have used thatν_x, ν_xx∈L^∞.

Finally, claim (d) is a direct consequence of claim (c).

Lemma 3.4. Letψbe aC¹function defined by(1.7). There exists a constantC >0such that for everyw₀∈ H, the solutionw of (3.10)–(3.11)satisfies

_T

0

ψ w(·, t)²_Hdt≥Cw0²_H. (3.12) Proof. By Duhamel, we know that there exists C >0 such that for w₀ ∈ H, the solution w of (3.10)–(3.11) satisfies

w0²_H≤C _T

0

w(·, t)²_Hdt. (3.13) Therefore, (3.12) will follow if we prove the following inequality inH:

_T

0

w(·, t)²_Hdt≤C _T

0

ψ w(·, t)²_Hdt. (3.14) We use the multiplier technique. Defineq∈C₀^∞(R)

q(x) =

xfor|x| ≤R+ 2

0 for |x| ≥R+ 3. (3.15)

We have that _T

0

d

dtw, iqwxdt=w, iqwx^T

0. (3.16)

ET AL.

RecallL=−∂_x²+α,then the l.h.s of the last equation reads:

_T

0

iw_xx, iqw_x − iαw+iP(w), iqw_x+w, iqiw_xxx+w, iq(−i)(αw+P(w))_xdt. (3.17) Integrating by parts we have that:

w, iqwx^T

0 = _T

0

−2wx, q_xw_x −2αw+P(w), qw_x − qxxw, w_x − αw+P(w), q_xwdt (3.18) and therefore, using thatf, g=Re

Rf g^∗: 1

2Im

Rqww¯_x^T

0 +Re _T

0

R

q_x|w_x|²+1

2q_xxww¯_x+ (αw+P(w))(qw¯_x+1 2q_xw)¯

dxdt= 0. (3.19) Then_T

0

|x|≤R+2|wx|²≤ ¹₂

{|x|≤R+3}qww¯_x^T

0

+_T

0

{R+2≤|x|≤R+3}q_x|wx|² + ¹₂

{R+2≤|x|≤R+3}q_xxww¯_x+

{|x|≤R+3}(αw+P(w))(qw¯_x+¹₂q_xw)¯ (3.20) and using Lemma3.3and

w(t₀, .)²_H1(R)≤C _T

0

w(t,·)²_H1(R)dt ∀t₀∈[0, T] (3.21)

αw_L²({|x|≤R+3})≤Cw_L²_(R) (3.22)

we have that there existε >0 and a constantC_ε such that _T

0

|x|≤R+2|wx|²dxdt≤ε _T

0

w(t,·)²_H1dt+C_{ε T}

0

w(t,·)²_L2dt (3.23) +C₂

_T

0

{R+2≤|x|≤R+3}|w_x|²dxdt. (3.24) We have that

wH≤ ψwH+(1−ψ)wH (3.25)

and since 1−ψ= 0 for|x|> R+ 1

(1−ψ)w_H ≤C(1−ψ)w_H¹. (3.26)

It is clear that

(1−ψ)w²_H1 ≤C

|x|≤R+1|wx|²dx+w²_L2(R)

, (3.27)

and since (ψw)_x=w_xfor|x| ≥R+ 2, we have that

|x|≥R+2|w_x|²dx≤ ψw²_H. (3.28) Therefore, ifεis chosen small enough, from (3.23) and (3.25)–(3.28), it follows the inequality

_T

0

w(·, t)²_Hdt≤C _T

0

ψ w(·, t)²_Hdt+ _T

0

w(·, t)²_L2dt

. (3.29)

It remains to prove that _T

0

w(·, t)²_L2dt≤C _T

0

ψw(·, t)²_Hdt. (3.30) Assume inequality (3.30) is not true, then there exists a sequencew₀^k∈ Hsuch that the corresponding sequence w^k of solutions of (3.10) satisfies

1 = _T

0

w^k(t)²_L2(R)dt≥k _T

0

ψw^k(t)²_Hdt, k= 1,2, . . . (3.31) According to (3.29) and (3.31), the sequence{w^k}is bounded in L²(0, T,H). Therefore by (3.13) the sequence {w₀^k} is bounded inH. Extracting a subsequence if needed, we may assume that

w^k₀ w₀ weakly inH and w^k w weakly inL²(0, T;H) (3.32) where w∈C([0, T];H) solves equation (3.10)–(3.11) with initial data w₀. Indeed, we first have that w₀^k w₀ weakly in Handw^k u weakly inL²(0, T,H). BeingHcompactly imbedded in L²(R), we may assume that w^k₀ →w₀ strongly in L²(R) and therefore

w^k→wstrongly inL²(0, T, L²(R)) (3.33)

where w ∈ C(0, T,H) since it is the solution of equation (3.10)–(3.11) with initial data w₀ ∈ H. From the uniqueness of weak limit inL²(0, T, L²(R)) we obtain thatw=u.

By (3.31), ψw^k →0 strongly in L²(0, T,H) and since ψw^k ψw weakly in L²(0, T,H), we conclude that ψw≡0 on R×(0, T). Consequently,

w(x, t) = 0,|x|> R+ 1, t∈(0, T). (3.34) Letv=L_μw, thenv satisfies equation (3.3) and

v(x, t) = 0,|x|> R+ 1, t∈(0, T). (3.35) We consider the new problem (similar to (3.3))

iv_t=−vxx+αψv˜ (3.36)

v(x,0) =v₀. where ˜ψ is aC₀^∞(R) given by

ψ(x) =˜

1 for|x| ≤R+ 1

0 for|x| ≥R+ 2. (3.37)

Then, problems (3.3) and (3.36) have the same solution which satisfy (3.35). Using Proposition 2.3 from [11]

with a=−αψ˜ andb = 0 functions in C₀^∞(R) and being v₀ ∈ H with compact support, we have that v is of classC^∞onR×(0, T).

By the unique continuation property for Schr¨odinger equation we conclude that v ≡ 0 on R×(0, T). This impliesw≡0 onR×(0, T). From (3.33) and (3.31) we have a contradiction.

Then observability inequality inH(3.12) is proved.

We are now in position to prove the observability inequality (3.9) inH. We first prove a weaker inequality:

Lemma 3.5. There exists a constant C > 0 such that for every v₀ ∈ H = W⁻¹ and v the solution of equation (3.3)–(3.4), the following inequality is satisfied

v0²_W−1≤C _T

0

ψv(t)²_W−1dt+v0²_W−2

. (3.38)

ET AL.

Proof. Suppose that inequality (3.38) is false. Then there exist a sequencev_k of solutions of (3.3) inC(0, T,H) such that

1 =v_k(0)²_W−1≥k _T

0

ψv_k(t)²_W−1dt+v_k(0)²_W−2

. (3.39)

Then we can extract a subsequence such that v_k(0) → v₀ weak in H for some v₀ ∈ H and we can assume v_k →0 strongly enW⁻²and thereforev₀= 0. Moreover, we can assumeψv_k →0 strongly inL²(0, T,H).

SinceH ⊂H¹(R) continuosly, we have that

w_xW⁰≤ w_W¹. (3.40)

Now, letv∈ H=W⁻¹, there existsw∈ H=W¹ such thatv=L_μw, then

v_xW⁻²=L_μw_x+μ_xw_W⁻² =L⁻¹_μ (L_μw_x+μ_xw)_W⁰ ≤ w_xW⁰+L⁻¹_μ μ_xw_W⁰ (3.41) using (3.40), we havev_xW⁻² ≤Cv_W⁻¹. From Lemma3.3we also know that there exists a constantC >0 such that for allw∈L²=W⁰

w_xW⁻¹≤Cw_W⁰. (3.42)

Next, we will prove thatv_k(0)→0 strongly inW⁻¹ arriving to a contradiction by (3.39).

Letw_k=L⁻¹_μ (v_k), thenw_k ∈C([0, T], W¹) is a solution of equation (3.10) inHand

ψw_k =ψL⁻¹_μ v_k=L⁻¹_μ (ψv_k) + [ψ, L⁻¹_μ ]v_k =L⁻¹_μ (ψv_k) +L⁻¹_μ [L_μ, ψ]w_k. (3.43) Sinceψv_k →0 strongly inL²(0, T,H) andL⁻¹_μ (ψv_k)1=ψv_k−1, we deduce thatL⁻¹_μ (ψv_k)→0 strongly in L²(0, T,H). On the other hand, using (3.42) and Lemma3.3we get

L⁻¹_μ [L_μ, ψ](w_k)_W¹ =[L_μ, ψ](w_k)_W⁻¹

=ψxxw_k+ 2ψ_x(w_k)_xW⁻¹

≤C(v_kW⁻³+v_kW⁻²)

≤Cv_kW⁻²

and this implies thatL⁻¹_μ [L_μ, ψ](w_k)→0 strongly inL²(0, T,H), sincev_k(0)→0 strongly inW⁻².

Thereforeψw_k →0 strongly inL²(0, T,H). Since w_k is a solution of (3.10) we have from the observability inequality (3.12) that w_k(0) → 0 strongly in H. It follows that v_k(0) = L_μw_k(0) → 0 strongly in H, which

contradicts the fact thatv_k(0)_H= 1.

Proof of Lemma 3.2. Assume that inequality (3.9) is false, then there exists a sequencev_k of solutions of (3.3) in C([0, T];H) such that

1 =vk(0)²_W−1 ≥k _T

0

ψvk(t)²_W−1dt (3.44)

for allk≥0.

Extracting a subsequence, we may assume that

v_k →v in L^∞(0, T;H) weak−,

v_k(0)→v(0) weakly in H (3.45)

for some solution v ∈ C(0, T;H) of (3.3)–(3.4). From (3.44), ψv_k → 0 strongly in L²(0, T,H) and since ψv_k→ψvin L^∞(0, T;H) weak-, we have thatψv≡0. We deduce as before thatv≡0 inR×(0, T).

{v_k(0)} being a bounded sequence in W⁻¹ and since W⁻¹ is compactly imbedded inW⁻², see (2.7), there exists a subsequence such thatv_k(0) converges strongly inW⁻² necessarily to 0.

We infer from (3.38) thatv_k(0) converges strongly to 0 in W⁻¹ which is absurd from (3.44). This finishes

the proof.