Stabilization of different evolution problems by internal dynamic controllers

Abstract

This thesis explores broadly the stabilization of complex dynamic systems by means of internal controllers. Specially, the asymptotic behavior of damped wave ? in an annular domain ? ? ?n, ?? ? 2, bounded by two edges ?0 and ?1 of class ??2. The inner edge ?0 is fasten so as to prevent any movement on this edge (Homogenous Dirichlet condition): ??|?0 = 0. As for the outer edge, the condition typically considered is: ??(??)?????? + ?????? ? ????? = 0, over ?1 × ?+, ?? ? ???(?1) where ?????? represents the acceleration of ?? (presence of kinetic energy), ?? ? 0 parameter allowing the presence\or absence of kinetic energy on ?1, ?? ? ?n normal vector pointing out of ? along ?1, ??? Laplace-Beltrami operator.

First of all, we start by the case where ?? > 0. In second position, we analyze the case where ?? = 0. For each case, we firstly demonstrate the existence and uniqueness of the solution by the semigroup method and the stability of each solution by analyzing the released energy over time. This energy analysis reveals a decrease in energy over time, thus suggesting a possible strong stability. For ?? > 0, we show the strong stability of the solution by exploiting the Arendt-Batty theorem combined with a unique continuation result. Then, we show the exponential stability for ?? > 0 (resp. ?? = 0) by a frequency approach of the ? domain (resp. the Nakao method). Finally, we conduct a series of numerical simulations to illustrate this exponential stability.