Approximation of exact controls for semi-linear 1D wave equations using a least-squares approach
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The exact distributed controllability of the semilinear wave equation y_tt-y_xx + g(y)=f 1_ω, 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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