Approximation of null controls for semilinear heat equations using a least-squares approach
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The null distributed controllability of the semilinear heat equation y_t-Δ y + g(y)=f 1_ω, assuming that g satisfies the growth condition g(s)/(| s|log^3/2(1+| s|))→ 0 as | s|→∞ and that g^'∈ L^∞_loc(ℝ) has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a fixed point argument makes use of precise estimates of the observability constant for a linearized heat equation. It does not provide however an explicit construction of a null control. Assuming that g^'∈ W^s,∞(ℝ) for one s∈ (0,1], 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, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In particular, after a finite number of iterations, the convergence is super linear with a rate equal to 1+s. Numerical experiments in the one dimensional setting support our analysis.
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