Global exact controllability in infinite time of Schrödinger equation

Nersesyan, Vahagn; Nersisyan, Hayk

doi:10.1016/j.matpur.2011.11.005

Cited by 30 publications

(26 citation statements)

References 30 publications

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“…The property of regularity is established in [2,Proposition 47]. In these references, the case of V = 0 is considered, but the case of a non-zero V is proved by literally the same arguments (see [19]). The time reversibility property is obvious.…”

Section: Well-posednessmentioning

confidence: 99%

Simultaneous global exact controllability of an arbitrary number of 1d bilinear Schrödinger equations

Morancey

Nersesyan

2015

Journal de Mathématiques Pures et Appliquées

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We consider a system of an arbitrary number of 1d linear Schrödinger equations on a bounded interval with bilinear control. We prove global exact controllability in large time of these N equations with a single control. This result is valid for an arbitrary potential with generic assumptions on the dipole moment of the considered particle. Thus, even in the case of a single particle, this result extends the available literature. The proof combines local exact controllability around finite sums of eigenstates, proved with Coron's return method, a global approximate controllability property, proved with Lyapunov strategy, and a compactness argument.

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Section: Well-posednessmentioning

confidence: 99%

Simultaneous global exact controllability of an arbitrary number of 1d bilinear Schrödinger equations

Morancey

Nersesyan

2015

Journal de Mathématiques Pures et Appliquées

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“…Other results have been obtained by Lyapunov-function techniques and combinations with local inversion results [8,28,34,35,36,38] and by geometric control methods, using Galerkin or adiabatic approximations [9,10,11,13,18,19]. Finally, let us conclude this necessarily incomplete list by mentioning that specific arguments have been developed to tackle physically relevant particular cases [6,20,30].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Controllability of Quantum Transport in an Electronic Nanostructure

Méhats¹,

Privat²,

Sigalotti³

2014

SIAM J. Appl. Math.

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We investigate the controllability of quantum electrons trapped in a twodimensional device, typically a MOS field-effect transistor. The problem is modeled by the Schrödinger equation in a bounded domain coupled to the Poisson equation for the electrical potential. The controller acts on the system through the boundary condition on the potential, on a part of the boundary modeling the gate. We prove that, generically with respect to the shape of the domain and boundary conditions on the gate, the device is controllable. We also consider control properties of a more realistic nonlinear version of the device, taking into account the self-consistent electrostatic Poisson potential.

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“…In this reference, the origin of the minimal time is the linearized system, whereas in the present article, the minimal time is related to the nonlinearity of the system. Exact controllability has also been studied in infinite time by Nersesyan and Nersisian in [28,29]. Now, we quote some approximate controllability results.…”

Section: A Review About Control Of Bilinear Systemsmentioning

confidence: 96%

Local controllability of 1D Schrödinger equations with bilinear control and minimal time

Beauchard

Morancey

2014

Mathematical Control &Amp; Related Fields

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We consider a linear Schrödinger equation, on a bounded interval, with bilinear control.In [10], Beauchard and Laurent prove that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. In [18], Coron proves that a positive minimal time is required for this controllability result, on a particular degenerate example.In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution.Theorem 2 There exists ǫ > 0 such that, for every d = 0 and u ∈ L 2 ((0, ǫ), R) satisfying |u(t)| < ǫ, ∀t ∈ (0, ǫ), the solution (ψ, s, d) ∈ C 0 ([0, ǫ], H 1 0 ((0, 1), C)) × C 0 ([0, ǫ], R) × C 1 ([0, ǫ], R) of (1.9) such that (ψ, s, d)(0) = (ψ 1 (0), 0, 0) satisfies (ψ, s, d)(ǫ) = (ψ 1 (ǫ), 0, d).The goal of this article is to go further in this analysis:1. we propose a general context for the minimal time to be positive (in particular, the variables s and d are not required anymore in the state), 2. we propose a sufficient condition for the local controllability to hold in large time; this assumption is compatible with the previous context and weaker than (1.7), 3. we work in an optimal functional frame, for instance, our non controllability result requires u small in L 2 -norm, not in L ∞ -norm as in Theorem 2, 4. we perform a first step toward the characterization of the minimal time.

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Global exact controllability in infinite time of Schrödinger equation

Cited by 30 publications

References 30 publications

Simultaneous global exact controllability of an arbitrary number of 1d bilinear Schrödinger equations

Simultaneous global exact controllability of an arbitrary number of 1d bilinear Schrödinger equations

On the Controllability of Quantum Transport in an Electronic Nanostructure

Local controllability of 1D Schrödinger equations with bilinear control and minimal time

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