Approximate Stabilization of One-dimensional Schrödinger Equations in Inhomogeneous Media

J Optim Theory Appl (2012) 153:758–768DOI 10.1007/s10957-011-9949-5

Approximate Stabilization of One-dimensionalSchrödinger Equations in Inhomogeneous Media

Jian Zu

Received: 27 September 2010 / Accepted: 21 October 2011 / Published online: 16 November 2011© Springer Science+Business Media, LLC 2011

Abstract We present how to control the bilinear 1D infinite-dimensional Schrödingerequations in inhomogeneous media (with x-dependent coefficients), getting the ap-proximate stabilization around ground state. Our arguments are based on constructinga Lyapunov function and a strategy similar to LaSalle invariance principle.

Keywords Lyapunov stabilization · LaSalle invariance principle · Bilinearx-dependent · Schrödinger equation · Inhomogeneous media

1 Introduction

The controllability of a finite-dimensional quantum system has been very well ex-plored [1–5]. There are mainly two classes of approaches commonly used by chemistsfor this task, optimal control techniques [6] and iterative stochastic techniques [7].

When some nondegeneracy assumptions, concerning the linearized system, aresatisfied, [8] provides another method based on Lyapunov techniques for generat-ing trajectories. The relevance of such a method for the control of chemical modelshas been studied in [9]. Implicit Lyapunov control of finite dimensional Schrödingerequations has been well solved in [10] by constructing a simple Lyapunov function.However, in infinite-dimensional case, trying to adapt the convergence analysis, basedon the use of the LaSalle invariance principle, the precompactness of the trajectoriesin L2 constitutes a major obstacle. In [11], the author gets the approximate stabiliza-tion of bilinear Schrödinger equation in infinite-dimensional systems, similar to [12],where the author prevents the population form going through the very high energy

Communicated by Viorel Barbu.

J. Zu (�)College of Mathematics, Jilin University, Changchun 130012, P.R. Chinae-mail: [email protected]

mailto:[email protected]

J Optim Theory Appl (2012) 153:758–768 759

levels, while trying to stabilize the system around the ground state. This work is animportant contribution to implicit feedback design.

It is quite relevant to consider this equation on inhomogeneous media [13]. Thex-dependent second order operator is fully discussed in [14] and [15]. In this paper,we consider the controllability of x-dependent Schrödinger equations and present theapproximate stabilization.

2 Preliminaries and Main Result

The symbol 〈·, ·〉 denotes the Hermitian product of L2([0,π],C):

⟨ψ(t, x), ξ(t, x)

⟩ :=∫ π

0u(x)ψ(t, x)ξ(t, x) dx.

We introduce the operator:

Aϕ := −1

2

(u(x)ϕx)x

u(x), D(A) := {

ϕ ∈ H 2([0,π],C) : ϕ(0) = ϕ(π) = 0

}.

It is well known that the eigenvectors of A construct an orthonormal basis (ϕk)k∈N ofL2([0,π],C):

Aϕk = λ2kϕk, k = 1,2, . . . ,

where (λk)k∈N is (increasingly) convergent to +∞ as k → +∞. For s > 0, we defineHs

(0)([0,π],C) := D(As/2), equipped with the norm:

‖η‖Hs(0)

:=( ∞∑

k=1

λ2sk |〈η,ϕk〉|2

)1/2

.

Now we consider a quantum particle on inhomogeneous media, with a potentialV (x), and coupled to an external(laser) field w(t) ∈ R through its dipole momentμ(x):

iψt = −1

2

(u(x)ψx)x

u(x)+ (

V (x) + w(t)μ(x))ψ.

Studying the infinite potential case:

V (x) = 0 for x ∈]0,π[ and V (x) = +∞ for x outside ]0,π[,then our system is

⎧⎪⎨

⎪⎩

iψt = − 12

(u(x)ψx)xu(x)

+ w(t)μ(x)ψ,

ψ(t,0) = ψ(t,π) = 0,

ψ(0, x) = ψ0(x).

(1)

It is a bilinear control system, in which

760 J Optim Theory Appl (2012) 153:758–768

– the state is the wave function ψ(t, x) : R+ ×R → C with ψ(t) ∈ S for every t ≥ 0,

and S := {ψ; ‖ψ‖L2[0,π] = 1},– u is x-dependent coefficient called the impedance function,– the control w : R

+ → R is an external interaction.

Denote a1 := 1, ak := 1 − ε(k ≥ 2), h′ := hx and{h(x)

}+ := max

{h(x),0

}.

Throughout this paper, we make the following assumptions:

Hypothesis

H1 We assume that μ ∈ W 2,∞[0,π], and for every j = k, 〈μϕj ,ϕk〉 = 0.

H2 u ∈ H 2(0,π);u(x) ≥ γ0 > 0,∀x ∈ [0,π] and

ρ = ess infηu(x) > −1

4, ρ1 = 2

π

∫ π

0

{ηu(x)

}+dx <

1

4, (2)

where ηu(x) = 12

u′′u

− 14 (u′

u)2.

Theorem 2.1 Under the Hypothesis H1,H2, for ε > 0, if there exists γ ∈]0,1[, 0 <

N ≤ 4, and ψ0 ∈ S ∩ H 2(0)[0,π] satisfying

∞∑

k=N+1

|〈ψ0, ϕk〉|2 <εγ 2

1 − εand |〈ψ0, ϕ1〉| ≥ γ, (3)

then the Cauchy problem (1) with w(ψ) = ς∑N

k=1 ak�(〈μψ,ϕk〉〈ϕk,ψ〉) has aunique strong solution. Furthermore, this solution satisfies

lim supt→+∞

distL2

(ψ(t), C1

) ≤ ε,

where Ck := {ϕkeiθ , θ ∈ [0,2π)}.

Remark 2.1 As 1 ≤ j < m ≤ 4, it admits a nondegenerate transition: λ2j − λ2

k =λ2

m − λ2n for (j, k) = (m,n). When u(x) = 1, (1) is a usual Schrödinger equation.

When N > 4, we can also consider this problem limiting |ηu(x)| small enough cor-responding to N by a perturbed method mentioned in [11, 16].

3 Properties of Linear Operators

In this section, we consider the Sturm–Liouville problem:

−(uϕ′

n

)x

= uλ2nϕn, ϕn(0) = ϕn(π) = 0, n ∈ N. (4)

By the substitution zn(x) = √u(x)ϕn(x), problem (5) is equivalent to the problem

z′′n(x) + (

λ2n − ηu(x)

)zn(x) = 0, zn(0) = zn(π) = 0. (5)


Lemma 3.1 [14, 17] Let ηu be a real function in L2[0,π] and let λ21 < λ2

2 < · · ·and z1, z2, . . . denote the eigenvalues and real eigenfunctions of the Sturm–Liouvilleproblem (5), then the following inequalities hold:

n2 + ρ ≤ λ2n ≤ λ + n2 + 2

π

∫ π

0

{λ − ηu(x)

}− dx (6)

for all λ < λ2n, n = 1,2, . . . , where ρ is given in (2) and {h(x)}− := {h(x)}+ − h(x).

Proof Firstly, we prove λ2n > ρ for all n = 1,2, . . .. Otherwise, ab absudo assume

that λ21 ≤ ρ, then λ2

1 ≤ ηu(x) for a.e. x ∈]0,π[. Multiplying (5) by z1 and integratingover ]0,π[, one concludes that z1 ≡ 0, which is absurd. Therefore, we have

ρ < λ21 < λ2

2 < · · · < λ2n < · · · .

Secondly, we obtain the lower bound for λ2n in (6). Indeed, zn has exactly n− 1 zeros

in ]0,π[, denoted by 0 = a0 < a1 < a2 < · · · < an−1 < an = π. In view of (5), wehave

(λ2

n − ρ)∫ ai

ai−1

z2n(x) dx =

∫ ai

ai−1

(z′n(x)

)2 + (ηu(x) − ρ

)z2n(x) dx,

∫ ai

ai−1

z2n(x) dx ≤ (ai − ai−1)

2

π2

∫ ai

ai−1

(z′n(x)

)2dx, i = 1, . . . , n,

which yields

∫ ai

ai−1

(z′n(x)

)2dx ≤ (λ2

n − ρ)(ai − ai−1)2

π2

∫ ai

ai−1

(z′n(x)

)2dx;

so, dividing by (ai − ai−1)2∫ ai

ai−1(z′

n(x))2dx and summing over i, we get

n∑

i=1

1

(ai − ai−1)2≤ (λ2

n − ρ)n

π2.

But min{∑ni=1

1x2i

; xi > 0, x1 + · · · + xn = π} = n( nπ)2 and it is assumed for x1 =

x2 = · · · = xn = πn, so

n2 ≤ λ2n − ρ.

Finally, we prove the upper bound for λ2n in (6). By the Prüfer transformation:

zn = r sin θ, z′n =

√λ2

n − λr cos θ,

with r(x) > 0, it is easy to obtain that

θ ′ =√

λ2n − λr cos2 θ − (ηu(x) − λ2

n) sin2 θ√

λ2n − λ


=√

λ2n − λ + (λ − ηu(x)) sin2(θ)

√λ2

n − λ

≥√

λ2n − λ − {λ − ηu(x)}− sin2(θ)

√λ2

n − λ

≥√

λ2n − λ − {λ − ηu(x)}−√

λ2n − λ

. (7)

We may take θ(a0) = 0, θ(an) = nπ . Integration of (7) over [0,π] yields that

nπ ≥ π

√λ2

n − λ −∫ π

0 {λ − ηu(x)}−dx√

λ2n − λ

,

which is equivalent to

D ≤ B√

D + C,

where D := λ2n − λ, B := n, and C := 1

π

∫ π

0 {λ − ηu(x)}−dx. Using the quadratic

formula of√

D, we get

√D ≤ [

B +√

B2 + 4C]/2. (8)

The elementary inequality√

1 + x ≤ 1 + x2 gives

√B2 + 4C = B

√

1 + 4C

B2≤ B

(1 + 2C

B2

)= B + 2C

B.

Squaring (8), we have

D ≤ [B2 + 2B√

B2 + 4C + B2 + 4C]4

≤ [B2 + 2B(B + 2CB

) + B2 + 4C]4

= B2 + 2C,

which is (6). The proof is complete. �

Lemma 3.2 Under the hypothesis H2, there exists a uniform gap |λ2j − λ2

k| > 1 for

any j = k. Furthermore, if 1 ≤ j,m ≤ 4 and k,n ∈ N, then λ2j − λ2

k = λ2m − λ2

n onlyfor (j, k) = (m,n).

Proof Obviously, for any 1 ≤ j < m ≤ 4, j = k, we have |m2 − j2 + k2 − n2| ≥1. Let λ = 0 in (6), we have n2 + ρ ≤ λ2

n ≤ n2 + 2π

∫ π

0 {η(x)}+dx. By H2, we getλ2

j − λ2k = λ2

m − λ2n for (j, k) = (m,n), 1 ≤ j < m ≤ 4, j = k. �


Now we prove the orthogonality of ϕn:

〈ϕn,ϕm〉 = 1

λ2n

⟨λ2

nϕn,ϕm

⟩ = 1

λ2n

∫ π

0−(

u(x)ϕ′n

)′ϕm dx

= 1

λ2n

∫ π

0ϕnλ

2mu(x)ϕmdx = λ2

m

λ2n

〈ϕn,ϕm〉. (9)

By Lemma 3.2, we get 〈ϕm,ϕn〉 = 0, when m = n. Obviously, ψk(t, x) := ϕk(x)e−iλkt

(k ∈ N) is a solution of (1) with w(t) = 0, and any solution can be divided intoψ(t, x) = ∑+∞

k=1 αkψk(t, x).

4 Approximate Stabilization

We denote that H−2 and H−1 are the dual space of H 2(0) and H 1

0 . The followingpropositions are similar to the propositions in [11].

Proposition 4.1 Let ψ0 ∈ S, T > 0, ω ∈ C([0, T ],R), and μ(x) ∈ W 2,∞[0,π].There exists a unique weak solution of (1), i.e., ψ ∈ C0([0, T ],S) ∩ C1([0, T ],H−2([0,π],C)):

ψ(t) = e−iAtψ0 − i

∫ t

0e−iA(t−s)w(s)μ(x)ψ(s) ds in L2([0,π],C),

and (1) holds in H−2([0,π]) for every t ∈ [0, T ]. Moveover, if ψ0 ∈ (H 2 ∩H 1

0 )([0,π],C), then ψ is a strong solution.

Proposition 4.2 Let (ψn0 )n∈N be a sequence of S, and let ψ∞

0 ∈ L2 with ‖ψ∞0 ‖L2 ≤

1, such that limt→+∞ ψn0 = ψ∞

0 strongly in H−1([0,π],C). Let ψn (resp., ψ∞) bethe weak solution of (1) with w(ψn) (resp., with w(ψ∞)), then for every τ > 0,limn→∞ ψn(τ) = ψ∞(τ ) strongly in H−1([0,π],C).

Proof By Poincaré inequality, there is a constant C′μ > 0 such that

‖μϕ‖H−1 ≤ C′μ‖ϕ‖H−1, for every ϕ ∈ H−1([0,π],C).

Considering

(ψn − ψ∞)

(t) = e−iAt(ψn

0 − ψ∞0

)

− i

∫ t

0e−iA(t−s)

[w

(ψn(s)

) − w(ψ∞(s)

)]μ(x)ψn(s) ds

− i

∫ t

0e−iA(t−s)w

(ψ∞(s)

)μ(x)

(ψn − ψ∞)

(s) ds.


Using w(ψ) in Theorem 2.1, ‖ψn(s)‖L2 = 1, ‖ψ∞(s)‖L2 ≤ 1 and the fact thatϕk, μ(x)ϕk(x) ∈ H 1

0 ([0,π],C) for k = 1, . . . ,N, we get∣∣w

(ψn(s)

) − w(ψ∞(s)

)∣∣ ≤ 2NC′μC(N)

∥∥(ψn − ψ∞)

(s)∥∥

H−1,

where C(N) := sup{‖ϕk‖H 10; k ∈ {1, . . . ,N}}. Because e−iAt preserves the H−1-

norm and |w(ψ∞(s)| ≤ N , it is easy to finish the proof thanks to the Gronwalllemma. �

Proof of Theorem 2.1 Let us construct the Lyapunov function:

Vε(ψ) = 1 − |〈ψ,ϕ1〉|2 − (1 − ε)

N∑

k=2

|〈ψ,ϕk〉|2 = 1 −N∑

k=1

ak〈ψ,ϕk〉〈ϕk,ψ〉. (10)

When ψ solves (1) with some control w, we have

dVε

dt= 2

N∑

k=1

ak�(〈ψt ,ϕk〉〈ϕk,ψ〉)

= 2N∑

k=1

ak�(⟨

(u(x)ψx)x

u(x),ϕk

⟩〈ϕk,ψ〉

)− 2w(t)

N∑

k=1

ak�(〈μψ,ϕk〉〈ϕk,ψ〉).

(11)

Since⟨(u(x)ψx)x

u(x),ϕk

⟩= u(x)ψxϕk |π0 −⟨

ψx, ϕ′k

⟩

= −u(x)ψϕ′k |π0 +

⟨ψ,

(u(x)ϕ′k)x

u(x)

⟩

= λ2k〈ψ,ϕk〉, (12)

we have

2N∑

k=1

ak�(⟨

(u(x)ψx)x

u(x),ϕk

⟩〈ϕk,ψ〉

)= 2

N∑

k=1

akλ2k�(〈ψ,ϕk〉〈ϕk,ψ〉) = 0,

and

dVε

dt= −2w(t)

N∑

k=1

ak�(〈μψ,ϕk〉〈ϕk,ψ〉).

Define the feedback law as

w(ψ(t)) := ς

N∑

k=1

ak�(〈μψ,ϕk〉〈ϕk,ψ〉),


where ς > 0. Obviously,

d

dtVε(ψ) = −2ςw(ψ(t))2 ≤ 0,

then Vε(ψ) is a nonincreasing function. Using ψ0 ∈ S and (3), we have

Vε(ψ0) = 1 − |〈ψ0, ϕ1〉|2 − (1 − ε)

N∑

k=2

|〈ψ0, ϕk〉|2

= 1 − ε|〈ψ0, ϕ1〉|2 − (1 − ε)

N∑

k=1

|〈ψ0, ϕk〉|2

< 1 − εγ 2 − (1 − ε)

(1 − εγ 2

1 − ε

)

< ε,

i.e.,

Vε(ψ) ≤ Vε(ψ0) < ε for all ψ.

From the definition of Vε(ψ), we know Vε(ψ) ≥ 0. Now we use a method similarto LaSalle invariance principle [18]. The trajectories of the closed-loop system con-verges to the largest invariant set contained in dVε

dt= 0. Let us determine this invariant

set. There exists a constant α ∈ [0, Vε(ψ0)] such that Vε(ψ(t)) → α, when t → +∞.This implies that

w(ψ) := ς

N∑

k=1

ak�(〈μψ,ϕk〉〈ϕk,ψ〉) → 0.

Let (tn)n∈N be an increasing sequence of positive real numbers such that tn → +∞.Since ‖ψ(tn)‖L2 = 1 for every n ∈ N, there exists a subsequence denoted again byψ(tn) ∈ L2 weakly convergence to ψ∞ ∈ L2([0,π],C) where ‖ψ∞‖L2 ≤ 1. Con-sider

⎧⎪⎨

⎪⎩

iξt = Aξ + w(ξ(t))μξ, x ∈ [0,π], t ∈ [0,+∞[,ξ(t,0) = ξ(t,π) = 0,

ξ(0) = ψ∞.

(13)

Thanks to Proposition 4.2,

ψ(tn + τ) → ξ(τ ) strongly in H−1([0,π],C), for every τ > 0.

Because Vε(.) is continuous for the L2-weak topology, we have

Vε(ψ(tn + τ)) → Vε(ξ(τ )) ≡ α.


Thus, ξ is a solution of (13) such that

dVε

dt= 0, i.e., w(ξ(τ )) ≡ 0.

Therefore, we have

ξ(τ ) =∞∑

k=1

〈ψ∞, ϕk〉ϕke−iλ2

kτ .

The equality w(ξ) = 0 gives

�(

N∑

k=1

∑

j∈N,j =k

ak〈ψ∞, ϕj 〉〈μϕj ,ϕk〉〈ψ∞, ϕk〉ei(λ2k−λ2

j )τ

)

≡ 0. (14)

Since ‖ψ∞‖L2 ≤ 1, we have

Vε(ψ∞) ≥ 1 − |〈ψ∞, ϕ1〉|2 − (1 − ε)

∞∑

k=2

|〈ψ∞, ϕk〉|2

= 1 − ε|〈ψ∞, ϕ1〉|2 − (1 − ε)[‖ψ∞‖2

L2 − |〈ψ∞, ϕ1〉|2]

≥ ε − ε|〈ψ∞, ϕ1〉|2.

Moreover, Vε(ψ∞) = α < ε, and thus |〈ψ∞, ϕ1〉|2 > 0. By Lemma 3.2, for T > 0large enough, using Ingham inequality (see [19], Theorem 1.2.9), the equality (14)implies, in particular,

〈ψ∞, ϕj 〉〈μ(x)ϕj ,ϕ1〉〈ϕ∞, ϕ1〉 = 0, ∀j ≥ 2.

Thanks to (H1), we get

〈ψ∞, ϕj 〉〈ϕ∞, ϕ1〉 = 0, ∀j ≥ 2,

i.e.,

〈ψ∞, ϕj 〉 = 0, ∀j ≥ 2.

Accordingly, there exists β ∈ C with |β| ≤ 1 such that ψ∞ = βϕ1, then

ε > α = Vε(ψ∞) = 1 − |β|2,

i.e.,

lim supt→∞

distL2(ψ(t), C1) ≤ ε.

The proof of the theorem is completed. �


Example 4.1 Considering system:

⎧⎪⎨

⎪⎩

iψt = − 12

((1+0.1 sinx)ψx)x(1+0.1 sinx)

+ w(t)μ(x)ψ,

ψ(t,0) = ψ(t,π) = 0,

ψ0 =√

22 ϕ1 +

√4900100 ϕ2 + 1

100ϕ7,

(15)

where μ satisfies the general assumption H1. For ε = 0.01 > 0, there exists γ =√2

2 > 0 satisfying all the assumptions of Theorem 2.1, which admits an approximatestabilization around ground state.

5 Conclusions

The approximate stabilization for a class of Schrödinger equations in inhomoge-neous media has been proved provided that it satisfies some assumptions. The generalSchrödinger equation is a special case in our paper.

Acknowledgements The author expresses his sincere thanks to Professor Yong Li and ProfessorShuguan Ji for useful discussions and invaluable suggestions. The author also thanks the referee for carefulreading of the manuscript and valuable comments.

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Approximate Stabilization of One-dimensional Schrödinger Equations in Inhomogeneous Media