Lp-asymptotic stability of 1D damped wave equations with localized and linear damping

Meryem Kafnemer (1) , Mebkhout Benmiloud (2) , Frédéric Jean (1) , Yacine Chitour

1 OC - Optimisation et commande
2 Département de Mathematiques [Tlemcen]

In this paper, we study the $L^p$-asymptotic stability of the one-dimensional linear damped wave equation with Dirichlet boundary conditions in $[0,1]$, with $p\in (1,\infty)$. The damping term is assumed to be linear and localized to an arbitrary open sub-interval of $[0,1]$. We prove that the semi-group $(S_p(t))_{t\geq 0}$ associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether $p\geq 2$ or $1

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

Meryem Kafnemer, Mebkhout Benmiloud, Frédéric Jean, Yacine Chitour. Lp-asymptotic stability of 1D damped wave equations with localized and linear damping. ESAIM: Control, Optimisation and Calculus of Variations, 2022, ⟨10.1051/cocv/2021107⟩. ⟨hal-03196874⟩