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

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