An observability estimate for the wave equation and applications to the Neumann boundary controllability for semi-linear wave equations

2024/09/11 by Sue Claret, Claret, Sue
Engineering · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Model Reduction and Neural Networks #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2409.07214

openalex publication_date 2024/09/11 · openalex created_date 2024/09/16 · openalex updated_date 2026/07/28

Abstract

We give a boundary observability result for a 1d wave equation with a potential. We then deduce with a Schauder fixed-point argument the existence of a Neumann boundary control for a semi-linear wave equation ∂_tty - ∂_xxy + f(y) = 0 under an optimal growth assumption at infinity on f of the type sln²s. Moreover, assuming additional assumption on f', we construct a minimizing sequence which converges to a control. Numerical experiments illustrate the results. This work extends to the Neumann boundary control case the work of Zuazua in 1993 and the work of Münch and Trélat in 2022.

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