Stabilization of systems and Ingham-Beurling inequalities

The main idea of this thesis is to provide the stabilization and the controllability up to renormalization of linear dynamics systems with unbounded feedback . That is mean the exponential stability and the controllability holds in a modified space defined via a natural weight function. As the result we will exprime the exponential decay of the renormalized energy by an observability inequality type. The second axis is devoted to study the pointwise observability, controllability and exponential stabilization of vibrating systems. In order to establish satisfactory stabilization theorems we will introduce functions spaces depending on the arithmetical properties of the stabilization point. Working in this framework for vibrating strings, beams and also for a coupled string-beam system; as a result we will construct a pointwise feedbacks leading to arbitrarily large prescribed decay rates. Finally, we will prove new results concerning the vectorial Ingham-Beurling theorem.