Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping

Khapalov, Alexander Y.

doi:10.1051/cocv:2006001

Cited by 11 publications

(7 citation statements)

References 15 publications

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“…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. beside the control v(x, t), a time-dependent control has been considered in the damped part.…”

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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“…We have seen in the proof of Theorem 4 that the image R T of the linear map dΘ T (0) is contained in the vector spacẽ (31). In order to prove Proposition 7, it is sufficient to prove that the image of the quadratic form d 2 Θ T (0) is not contained inR T .…”

Section: Propositionmentioning

confidence: 97%

“…Such controllability problems may arise in the context of 'smart materials', whose properties can be altered by applying various factors (temperature, electric current, magnetic field). In [31], the author proves the global approximate controllability to nonnegative equilibrium states: ∀(y 0 ,…”

Section: Wave Equation With Bilinear Controlmentioning

confidence: 99%

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

Beauchard

2011

Journal of Differential Equations

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We consider a linear wave equation, on the interval (0, 1), with bilinear control and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory. We prove that the following results hold generically.• For every T > 2, this system is locally controllable in H 3 × H 2 , in time T , with controls in L 2 ((0, T ), R). • For T = 2, this system is locally controllable up to codimension one in H 3 × H 2 , in time T , with controls in L 2 ((0, T ), R): the reachable set is (locally) a non-flat submanifold of H 3 × H 2 with codimension one.• For every T < 2, this system is not locally controllable, more precisely, the reachable set, with controls in L 2 ((0, T ), R), is contained in a non-flat submanifold of H 3 × H 2 , with infinite codimension.The proof of these results relies on the inverse mapping theorem and second order expansions.

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“…Finally, we quote some references concerning bilinear wave equations. In [24,23,22], Khapalov considers nonlinear wave equations with bilinear controls. He proves the global approximate controllability to nonnegative equilibrium states.…”

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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Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping

Cited by 11 publications

References 15 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 and non-controllability for a 1D wave equation with bilinear control

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

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