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

2020/10/25 by Münch, Arnaud, Trélat, Emmanuel
#35Q30 #93E24 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2010.14067

Abstract

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 \vert g(s)\vert /(\vert s\vert log²(\vert s\vert))→ 0 as \vert s\vert → ∞ 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} \vert g^′(a)-g^′(b)\vert/\vert a-b\vert^r

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