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Pré-Publication, Document De Travail Année : 2022

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

Résumé

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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hal-03599036 , version 1 (06-03-2022)

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  • HAL Id : hal-03599036 , version 1

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Kuntal Bhandari, Jérôme Lemoine, Arnaud Munch. Exact boundary controllability of 1D semilinear wave equations through a constructive approach. 2022. ⟨hal-03599036⟩
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