The purpose of this talk is to give an overview of recent progresses for stabilization of continuous and discretized PDE's. Several results and methods have been developed in this direction, however they do not necess arily lead to one-shoot, simple and explicit energy decay formulas. Moreover, several of these results do not give the expected "optimal" decay rates.
This talk will present a general method, the "optimal-weight convexity method" (see [4, 1, 2] for frictional dampings and [8,9] for memory dampings ) which provides one-shoot, simple and explicit energy decay formulas for nonlinear frictional dampings as well as memory ones.
Moreover, these results are valid under optimal geometric conditions using indirect methods through observability estimates for the undamped problem.
We then discuss optimality of energy decay rates both in the finite and infinite dimensional cases. These optimal estimates are based on an energy comparison principle and comparison between pointwise and energy estimates. In the infi nite dimensional case, optimality or lower energy estimates have been obtained only in a few, speci c cases. We shall give a methodology based on the dissipation relation and interpolation properties to derive lower energy estimates for several new problems and indicate the open problems (see [2, 5, 6]).
We also present further results on the discretized PDE's ([7]) and results on coupled systems [10].