Local controllability and non-controllability for a 1D wave equation with bilinear control

Beauchard, Karine

doi:10.1016/j.jde.2010.10.008

Cited by 39 publications

(28 citation statements)

References 39 publications

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“…Various approaches were used to tackle the question of multiplicative controllability of hyperbolic equations like (1). The homogeneous version of (1) (i.e, f = 0) has been considered in [3,8,19,21,30]. The case of semilinear wave equation has been studied in [20] for equilibriumlike states of the form (y d 1 , 0) using two controls, i.e.…”

Section: Introductionmentioning

confidence: 99%

Controllability of the semilinear wave equation governed by a multiplicative control

Ouzahra¹

2019

Evolution Equations &Amp; Control Theory

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In this paper we establish several results on approximate controllability of a semilinear wave equation by making use of a single multiplicative control. These results are then applied to discuss the exact controllability properties for the one dimensional version of the system at hand. The proof relies on linear semigroup theory and the results on the additive controllability of the semilinear wave equation. The approaches are constructive and provide explicit steering controls. Moreover, in the context of undamped wave equation, the exact controllability is established for a time which is uniform for all initial states.

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Section: Introductionmentioning

confidence: 99%

Controllability of the semilinear wave equation governed by a multiplicative control

Ouzahra¹

2019

Evolution Equations &Amp; Control Theory

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“…In [18], Coron proved that a positive minimal time may be required for the local controllability of the 1D model. In [8], Beauchard studied the minimal time for the local controllability of 1D wave equations with bilinear controls. 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.…”

Section: A Review About Control Of Bilinear Systemsmentioning

confidence: 99%

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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“…The study of a toy model (Beauchard 2011) suggests that if local controllability holds in 2D (with a priori bounded L 2 -controls) then a positive minimal time is required, whatever is. The appropriate functional frame for such a result is an open problem.…”

Section: Open Problems In Multi-d or With Continuous Spectrummentioning

confidence: 99%

Bilinear Control of Schrödinger PDEs

Beauchard¹,

Rouchon²

2013

Encyclopedia of Systems and Control

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This entry is an introduction to modern issues about controllability of Schrödinger PDEs with bilinear controls. This model is pertinent for a quantum particle, controlled by an electric field. We review recent developments in the field, with discrimination between exact and approximate controllabilities, in finite or infinite time. We also underline the variety of mathematical tools used by various teams in the last decade. The results are illustrated on several classical examples.

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Local controllability and non-controllability for a 1D wave equation with bilinear control

Cited by 39 publications

References 39 publications

Controllability of the semilinear wave equation governed by a multiplicative control

Controllability of the semilinear wave equation governed by a multiplicative control

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

Bilinear Control of Schrödinger PDEs

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