Stabilization of the Wave Equation by the Mean of a Saturating Dirichlet Feedback

Nicolas Vanspranghe (1) , Francesco Ferrante (1) , Christophe Prieur (1)

1 GIPSA-INFINITY - GIPSA - Infinite Dimensional Dynamics

In this paper, we consider the wave equation with Dirichlet boundary control subject to a nonlinearity, the kind of which includes (but is not restricted to) pointwise saturation mappings. The case where only a subset of the boundary is actuated is allowed. Initial data is taken in the optimal energy space associated with Dirichlet boundary control-which means that we deal with (very) weak solutions. Using nonlinear semigroup techniques, we prove that the associated closed-loop system is asymptotically stable. Some numerical simulations are given to illustrate the stability result.

Mots clés: Infinite-dimensional systems; stability of nonlinear systems; control of partial differential equations; control and estimation of wave equations and systems of elasticity; semigroup and operator theory

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

Nicolas Vanspranghe, Francesco Ferrante, Christophe Prieur. Stabilization of the Wave Equation by the Mean of a Saturating Dirichlet Feedback. MICNON 2021 - Third IFAC Conference on Modelling, Identification and Control of Nonlinear Systems, Sep 2021, Tokyo, Japan. ⟨10.1016/j.ifacol.2021.10.394⟩. ⟨hal-03349953⟩