$L^p$-asymptotic stability analysis of a 1D wave equation with a boundary nonmonotone damping

Swann Marx (1) , Guilherme Mazanti (2, 3)

1 LS2N - Laboratoire des Sciences du Numérique de Nantes
2 IPSA - Institut Polytechnique des Sciences Avancées
3 DISCO - Dynamical Interconnected Systems in COmplex Environments

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with a nonlinear non-monotone damping acting at a boundary. The study is performed in an $L^p$-functional framework, $p\in [1,\infty]$. Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the decay rate. The well-posedness results rely mainly on some results collected in [7]. Asymptotic behavior results are obtained by the use of a suitable Lyapunov functional if $p$ is finite and on a trajectory-based analysis if $p=\infty$.

Mots clés: Wave equation; boundary control; nonmonotone damping; nonlinear damping; asymptotic stability; exponential stability

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Swann Marx, Guilherme Mazanti. $L^p$-asymptotic stability analysis of a 1D wave equation with a boundary nonmonotone damping. 2020. ⟨hal-03303057v1⟩