The purpose of this paper is to discuss an output stabilization problem for linear infinite dimensional systems-regional stabilizability-and consists in studying the asymptotic behaviour of a distributed system on a subregion of its evolution domain. So we give definitions and some properties of this kind of stability. Also, we concentrate on the determination of the control which ensures regional stabilizability on a subregion interior to the system domain, with bounded state on the residual part and minimizing a given cost of performance. The obtained results are illustrated by many examples and simulations.