Explicit approximate controllability of the Schrödinger equation with a polarizability term
We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts non linearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schrödinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system with explicit controls. Numerical simulations are presented to illustrate those theoretical results.
Key words : Approximate controllability, Schrödinger equation, polarizability, oscillating controls, averaging, feedback stabilization, LaSalle invariance principle.
1.1 Main result
Following [RouchonModele] we consider a quantum particle in a potential and an electric field of amplitude . We assume that the dipolar approximation is not valid (see [Dion_1, Dion_2]). Then, the particle is represented by its wave function solution of the following Schrödinger equation
with initial condition
where is a bounded domain with smooth boundary. The functions are given, is the dipolar moment and the polarizability moment. For the sake of simplicity, we denote by , and respectively the usual Lebesgue and Sobolev spaces , and . The following well-posedness result holds (see [Caz]) by application of the Banach fixed point theorem.
Let and be the usual scalar product on
We consider the operator with domain and denote by the non decreasing sequence of its eigenvalues and by the associated eigenvectors in . The family is a Hilbert basis of .
Our goal is to stabilize the ground state. As the global phase of the wave function is physically meaningless, our target set is
Let and . We assume that the following hypotheses hold.
i.e. all coupling are realized either by or ,
i.e. only a finite number of coupling is missed by ,
for such that and ,
The hypothesis i) is weaker than the one in [BeauchardNersesyan] (i.e. ). As proved in [Nersesyan, Section 3.4], we get that generically with respect to and in , the scalar products and are all non zero. The spectral assumption iii) does not hold in every physical situation. For example, it is not satisfied in 1D if . However, it is proved in [Nersesyan, Lemma 3.12] that if is the rectangle , Hypothesis 1.1 iii) hold generically with respect to in the set .
As in [CGLT], we use a time-periodic oscillating control of the form
Following classical techniques (see e.g. [SandersVerhulst07]) of dynamical systems in finite dimension let us introduce the averaged system
with initial condition
Let be the orthogonal projection in onto the closure of Span and be a positive constant (to be determined later).
Our stabilization strategy relies on the following Lyapunov function (used in [BeauchardNersesyan]) defined on by
This leads to feedback laws given by
with small enough and
and for , for ,
We can now state the well-posedness of the averaged closed loop system (1.5).
We define the set of admissible initial conditions. For an initial condition , we define the control
The main result of this article is the following one.
Assume that Hypotheses 1.1 hold. Let , the target set, be defined by (1.3). There exists such that for any , for any and for any with , there exist an increasing time sequence in tending to and a decreasing sequence in such that if is the solution of (1.1)-(1.2) associated to the control defined by (1.12) then for all , if ,
Theorem 1.1 gives the semi-global approximate controllability with explicit controls of system (1.1). Hypotheses 1.1 are needed to ensure that the invariant set coincides with the target set. The semi-global aspect comes from the hypothesis : by reducing (in a way dependant of ), this condition can be fulfilled as soon as .
In Theorem 1.1, there is a gap between the regularity of the initial condition and the approximate controllability in with . The extra-regularity is used in this article to prove an approximation property in between the oscillating system and the averaged one (see Section 3). Weakening this regularity assumption is an open problem for which an alternative strategy is required. The last lost of regularity comes from the application of a weak LaSalle principle instead of a strong one due to lack of compactness in infinite dimension.
1.2 A review of previous results
In this section, we recall previous results about quantum systems with bilinear controls. The model (1.1) of an infinite potential well was proposed by Rouchon in [RouchonModele] in the dipolar approximation (). A classical negative result was obtained in [BallMarsdenSlemrod82] by Ball, Marsden and Slemrod for infinite dimensional bilinear control systems. This result implies, for system (1.1) with , that the set of reachable states from any initial data in with control in has a dense complement in . However, exact controllability was proved in 1D by Beauchard in [Beauchard05] for and in more regular spaces (). This result was then refined in [BeauchardLaurent] by Beauchard and Laurent for more general and a regularity .
The question of stabilization is addressed in [BeauchardNersesyan] where Beauchard and Nersesyan extended previous results from Nersesyan [Nersesyan]. They proved, under appropriate assumptions on , the semi-global weak stabilization of the wave function towards the ground state using explicit feedback control and Lyapunov techniques in infinite dimension.
However sometimes, for example in the case of higher laser intensities, this model is not efficient (see e.g. [Dion_1, Dion_2]) and we need to add a polarizability term in the model. This term, if not neglected, can also be helpful in mathematical proofs. Indeed the result of [BeauchardNersesyan] only holds if couples the ground state to any other eigenstate and then the use of the polarizability enables us to weaken this assumption. Mathematical use of the expansion of the Hamiltonian beyond the dipolar approximation was used by Grigoriu, Lefter and Turinici in [GrigoriuLefterTurinici09, Turinici07]. A finite dimension approximation of this model was studied in [CGLT] by Coron, Grigoriu, Lefter and Turinici. The authors proposed discontinuous feedback laws and periodic highly oscillating feedback laws to stabilize the ground state. In this article, we extend in our infinite dimensional framework their idea of using (time-dependent) periodic feedback laws. We also refer to the book [CoronBook] by Coron for a comprehensive presentation of the feedback strategy and the use of time-varying feedback laws.
How to adapt the Lyapunov or LaSalle strategy in an infinite dimensional framework is not clear because closed bounded sets are not compact so the trajectories may lack compactness in the considered topology. In this direction we should cite some related works of Mirrahimi and Beauchard [BeauchardMirrahimi09, Mirrahimi09] where the idea was to prove approximate convergence results. In this article, we will use an adaptation of the LaSalle invariance principle for weak convergence which was used for example in [BeauchardNersesyan] by Beauchard and Nersesyan. There are other strategies to show a strong stabilization property. Coron and d’Andréa-Novel proved in [CAN] the compactness of the trajectories by a direct method for a beam equation and thus the strong stabilization. In [Couchouron1, Couchouron2] Couchouron gave sufficient conditions to obtain the compactness in favorable cases where the control acts diagonally on the state. Another strategy to obtain strong results is to look for a strict Lyapunov function, which is an even trickier question, and was done for example in [CoronAndreaNovelBastin07] by Coron, d’Andréa-Novel and Bastin for a system of conservation laws.
The question of approximate controllability has been addressed by various authors using various techniques. In [Nersesyan10], Nersesyan uses a Lyapunov strategy to obtain approximate controllability in large time in regular spaces. In [CMSB09], Chambrion, Mason, Sigalotti and Boscain proved approximate controllability in for a wider class of systems using geometric control tools for the Galerkin approximations. The hypotheses needed were weakened in [BCCS11] and the approximate controllability was extended to some spaces in [BoussaidCaponigroChambrion].
Explicit approximate controllability in large time has also been obtained by Ervedoza and Puel in [ErvedozaPuel09] on a model of trapped ion, using different tools.
1.3 Structure of this article
As announced in Section 1.1, we study the system (1.1) by introducing a highly oscillating time-periodic control and the corresponding averaged system. Section 2 is devoted to the introduction of this averaged system and its weak stabilization using Lyapunov techniques and an adaptation of the LaSalle invariance principle in infinite dimension.
In Section 3 we study the approximation property between the solution of the averaged system and the solution of (1.1) with the same initial condition. We prove that on every finite time interval these two solutions remain arbitrarily close provided that the control is oscillating enough. This is an extension of classical averaging results for finite dimension dynamical systems.
2 Stabilization of the averaged system
2.1 Definition of the averaged system
where the operator is defined by , and
For any , is -periodic (with here ). Following classical techniques of averaging, we introduce . We can define the averaged system associated to (2.1) by
We show by Lyapunov techniques that we can choose and such that the solution of the averaged system (2.3) is weakly convergent in towards our target set .
2.2 Control Lyapunov function and damping feedback laws
Our candidate for the Lyapunov function, , is defined in (1.7). It is clear that whenever and that if and only if .
The main advantage of this Lyapunov function is that it can be used to bound the norm. In fact, for any ,
where here, as in all this article, is a positive constant possibly different each time it appears. This leads to the existence of satisfying
Although the idea of using a feedback of the form (1.4) is inspired by [CGLT], the construction of the Lyapunov function and of the controls is here different because we are dealing with an infinite dimensional framework. We follow the strategy used in [Nersesyan, BeauchardNersesyan].
Choice of the feedbacks.
If for all then
Then we perform integration by parts. As commutes with , is real and thanks to the following boundary conditions
This leads to
where is defined in (1.10).
Well-posedness and boundedness proofs.
Using the previous heuristic on the Lyapunov function, we can state and prove the well-posedness of the closed loop system (1.5)-(1.8) globally in time and derive a uniform bound on the norm of the solution. Namely, we prove Proposition 1.2.
Proof of Proposition 1.2..
By the explicit expression (1.10) of , we get for any , where
Notice that is increasing on . Let where is defined by (2.4), and .
The local existence and regularity is obtained by a classical fixed point argument : there exists such that the closed loop system (1.5) with initial condition (1.6) and feedback laws (1.8) admits a unique solution defined on and satisfying either or and
thus, by continuity, for small enough.
We want to prove that .
Let us proceed by contradiction and assume that . This implies . By definition of ,
This is inconsistent with so and the solution is bounded in when it is defined. As no blow-up is possible thanks to (2.8) we obtain that and thus the solution is global in time and bounded.
Finally, taking the time derivative of the equation we obtain the announced regularity. ∎
2.3 Convergence Analysis
We prove this theorem by adapting the LaSalle invariance principle to infinite dimension in the same spirit as in [BeauchardNersesyan]. This is done in two steps. First we prove that the invariant set, relatively to the closed-loop system (1.5)-(1.8) and the Lyapunov function , is . Here, Hypothesis 1.1 is crucial. Then we prove that every adherent point for the weak topology of the solution of this closed-loop system is contained in . This is due to the continuity of the propagator of the closed-loop system for the weak topology.
2.3.1 Invariant set
Assume that Hypotheses 1.1 hold. Assume that belongs to and satisfies . If the function is constant, then .
By (1.8) this implies that and then is solution of the uncontrolled Schrödinger equation. So,
Recall that is the ground state. Following the idea of [Nersesyan], we obtain after computations and gathering the terms with different exponential term
where and are constants. Then, by [Nersesyan, Lemma 3.10],
Using the assumption and Hypotheses 1.1 it comes that for all , . This leads to
where by Hypotheses 1.1, is a finite set. By simple computations we obtain,
There exists and such that
with and if . Thus, (2.8) implies that for any , there exists such that
Straightforward computations give
Thus, and our inequality (2.10) can be rewritten as
with the being all different and non-zero. Then using the same argument as in [CGLT, Proof of Theorem 3.1], we get that for and then using (2.8) in particular that the coefficient of vanishes. It implies for all . Consequently, . As , we obtain . ∎
2.3.2 Weak continuity of the propagator
We denote by the propagator of the closed-loop system (1.5)-(1.8). We detail here the continuity property of this propagator and of the feedback laws we need to apply the LaSalle invariance principle.
Let be a sequence such that in . For every , there exists of zero Lebesgue measure verifying for all ,
Proof of ii).
We start by proving that if satisfy
in then and . Thus will be a simple consequence of . As proved in [BeauchardNersesyan, Proposition 2.2], using the fact that the regularity is sufficient to define the feedback, we get
So by the design of our feedback,
Proof of i). The exact same proof as in [BeauchardNersesyan, Proposition 2.2] based on extraction in less regular spaces, uniqueness property of the closed loop system and taking into account the polarizability term leads to the announced result. ∎
2.3.3 LaSalle invariance principle
We now have all the needed tools to prove Theorem 2.1.
Proof of Theorem 2.1.
Consider with . Thanks to the bound (2.4), is bounded in . Let be a sequence of times tending to and be such that in . We want to show that .
We prove that and . Indeed, the function belongs to (because of (2.6) and (1.8)) so the sequence of functions tends to zero in . Then by the Lebesgue reciprocal theorem there exists a subsequence and of zero Lebesgue measure such that
Let . Using Proposition 2.2, there exists of zero Lebesgue measure such that
Hence, for all . The function being continuous we get for all , and this for all . Finally for all .
The same argument holds for as belongs to . Then by the proof of Proposition 2.2,
and implies .
3 Approximation by averaging
The method of averaging was mostly used for finite-dimensional dynamical systems (see e.g. [SandersVerhulst07]). The concept of averaging in quantum control theory has already produced interesting results. For example, in [MirrahimiSarletteRouchon10] the authors make important use of these averaging properties in finite dimension through what is called in quantum physics the rotating wave approximation. The main idea of using a highly oscillating control is that if it is oscillating enough the initial system behaves like the averaged system. We extend this concept in our infinite dimensional framework : we prove an approximation result on every finite time interval. More precisely we have the following result.
Notice that the controls and were defined using the averaged system in a feedback form but the control used for the system (1.1) is explicit and is not defined as a feedback control.
Due to the infinite dimensional framework, we are facing regularity issues and cannot adapt directly the strategy of [SandersVerhulst07].
We define for ,
Notice that thanks to (2.2) for any ,