Local controllability of a 1-D Schrödinger equation

Abstract

We consider a non relativistic charged particle in a 1-D box of potential. This quantum system is subject to a control, which is a uniform electric field. It is represented by a complex probability amplitude solution of a Schrödinger equation. We prove the local controllability of this nonlinear system around the ground state. Our proof uses the return method, a Nash-Moser implicit function theorem and moment theory.Résumé: On considère une particule non relativiste dans un puits de potentiel en dimension un d'espace. Ce systeme quantique est soumisà un champélectrique uniforme, qui constitue un contrôle. Il est représenté par une densité de probabilité complexe solution d'uneéquation de Schrödinger. On démontre la contrôlabilité locale de ce système non linéaire au voisinage de l'état fondamental. La preuve utilise la méthode du retour, un théorème de Nash-Moser et la théorie des moments.

Implicit Lyapunov control of finite dimensional Schrödinger equations

“…A test case to be looked is the control of a quantum particle in a moving potential well. It is proved in [20] that the first order approximation for this system is not controllable and in [3,4] that the nonlinear system is locally controllable around any eigenstate (we are in the same situation as in the 5-level system of the last section). It seems that, for such a system, the assumptions of Theorem 3 are fulfilled.…”

Limitations on the control of Schrödinger equations

“…It is also interesting to compare Theorem 6 (in the linear case f ≡ 0) to the results in [8,9], which show that (local) controllability is recovered if the quadratic potential is replaced with an "infinite" confining potential ("particle in a box"). This illustrates the subtle dependence of the controllability properties on the external potential.…”

“…to the space of controls; controllability may be recovered by working in a different (higher-regularity) state space. This phenomenon occurs in two recent papers by Beauchard and Coron [8,9]. The bilinear control problem considered there falls in the scope of the noncontrollabilty result by Turinici if the state space is chosen to be H 2 .…”

“…Results on distributed and/or boundary control for linear Schrödinger equations are the subjects of references [15, 25, 27-29, 33, 34, 36]; local exact controllability for the linear Schrödinger equation with bilinear control is shown in [8,9]. A small-data result on distributed exact control for the nonlinear Schrödinger equation was presented in [20] (for completeness, this result is listed in Sect.…”

Ensemble controllability and discrimination of perturbed bilinear control systems on connected, simple, compact Lie groups

“…Among recent applications one may cite the field of quantum control with optical or magnetic external fields (see [5,[9][10][11][12][13][14][15][16][17][18][19]). …”