Numerical Control of a Semilinear Wave Equation on an Interval

Cavalcanti, Marcelo M.; Rosier, Carole; Rosier, Lionel

doi:10.1007/978-3-030-94766-8_4

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…As we shall see, this is related to an appropriate choice of the parameter s related to the norm of the initial condition. Again, to our knowledge, this is the first result leading to a convergent approximation of boundary controls for superlinear nonlinearities without smallness assumption notably on the initial condition and target (contrary to the recent works [CCCR21] and [NZWF19]).…”

Section: < +∞mentioning

confidence: 55%

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Exact boundary controllability of 1D semilinear wave equations through a constructive approach

Bhandari

Lemoine

Münch

2022

Math. Control Signals Syst.

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The exact boundary controllability of the semilinear wave equation ytt − yxx + f (y) = 0, x ∈ (0, 1) assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup |r|→∞ |f (r)|/(|r| ln p |r|) β for some β small enough and p = 2 has been obtained by Zuazua in 1993. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized wave equation. Under the above asymptotic assumption with p = 3/2, by introducing a different fixed point application, we present a simpler proof of the exact boundary controllability which is not based on the cost of observability of the wave equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition lim sup |r|→∞ |f (r)|/ ln 3/2 |r| β for some β small enough, we show that the above fixed point application is contracting yielding a constructive method to approximate the controls for the semilinear equation. Numerical experiments illustrate the results. The results can be extended to the multi-dimensional case and for nonlinearities involving the gradient of the solution.

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Section: < +∞mentioning

confidence: 55%

“…To our knowledge, this result is the first proposing a constructive approximation of boundary controls for the semilinear wave equation without the assumption that f is globally Lipschitz. Under smallness assumptions on the data, we mention the recent works [CCCR21] and [NZWF19].…”

Section: < +∞mentioning

confidence: 99%

Exact boundary controllability of 1D semilinear wave equations through a constructive approach

Bhandari

Lemoine

Münch

2022

Math. Control Signals Syst.

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Control of Partial Differential Equations via Physics-Informed Neural Networks

Garcı́a-Cervera

Kessler

Periago

2022

J Optim Theory Appl

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This paper addresses the numerical resolution of controllability problems for partial differential equations (PDEs) by using physics-informed neural networks. Error estimates for the generalization error for both state and control are derived from classical observability inequalities and energy estimates for the considered PDE. These error bounds, that apply to any exact controllable linear system of PDEs and in any dimension, provide a rigorous justification for the use of neural networks in this field. Preliminary numerical simulation results for three different types of PDEs are carried out to illustrate the performance of the proposed methodology.

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