Stabilization of semilinear PDE's, and uniform decay under discretization

Abstract

These notes are issued from a short course given by the author in a summer school in Chamb{\'e}ry in June 2015. We consider general semilinear PDE's and we address the following two questions: 1) How to design an efficient feedback control locally stabilizing the equation asymptotically to 0? 2) How to construct such a stabilizing feedback from approximation schemes? To address these issues, we distinguish between parabolic and hyperbolic semilinear PDE's. By parabolic, we mean that the linear operator underlying the system generates an analytic semi-group. By hyperbolic, we mean that this operator is skew-adjoint. We first recall general results allowing one to consider the nonlinear term as a perturbation that can be absorbed when one is able to construct a Lyapunov function for the linear part. We recall in particular some known results borrowed from the Riccati theory. However, since the numerical implementation of Riccati operators is computationally demanding, we focus then on the question of being able to design "simple" feedbacks. For parabolic equations, we describe a method consisting of designing a stabilizing feedback, based on a small finite-dimensional (spectral) approximation of the whole system. For hyperbolic equations, we focus on simple linear or nonlinear feedbacks and we investigate the question of obtaining sharp decay results. When considering discretization schemes, the decay obtained in the continuous model cannot in general be preserved for the discrete model, and we address the question of adding appropriate viscosity terms in the numerical scheme, in order to recover a uniform decay. We consider space, time and then full discretizations and we report in particular on the most recent results obtained in the literature. Finally, we describe several open problems and issues.