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Approximation of exact controls for semi-linear 1D wave equations using a least-squares approach

Abstract : The exact distributed controllability of the semilinear wave equation ytt − yxx + g(y) = f 1ω, assuming that g satisfies the growth condition |g(s)|/(|s| log 2 (|s|)) → 0 as |s| → ∞ and that g ∈ L ∞ loc (R) has been obtained by Zuazua in the nineties. The proof based on a Leray-Schauder fixed point argument makes use of precise estimates of the observability constant for a linearized wave equation. It does not provide however an explicit construction of a null control. Assuming that g ∈ L ∞ loc (R), that sup a,b∈R,a =b |g (a) − g (b)|/|a − b| r < ∞ for some r ∈ (0, 1] and that g satisfies the growth condition |g (s)|/ log 2 (|s|) → 0 as |s| → ∞, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach guarantees the convergence whatever the initial element of the sequence may be. In particular, after a finite number of iterations, the convergence is super linear with rate 1 + r. This general method provides a constructive proof of the exact controllability for the semilinear wave equation.
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https://hal.archives-ouvertes.fr/hal-02975370
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Submitted on : Thursday, October 22, 2020 - 4:07:14 PM
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  • HAL Id : hal-02975370, version 1

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Arnaud Munch, Emmanuel Trélat. Approximation of exact controls for semi-linear 1D wave equations using a least-squares approach. 2020. ⟨hal-02975370v1⟩

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