In this paper, a link between Poly-Quadratic stability and stability of a class of Hybrid systems is established. This class concerns the class of linear switched systems. Poly-Quadratic stability aims to check asymptotic stability of a polytopic system by mean of polytopic quadratic Lyapunov functions. The necessary and sufficient condition of Poly-Quadratic stability proposed previously is shown to be immediately applicable to a class of switched control and observer design problems. Chaos synchronization for which the transmitter is described by a piecewise linear map is presented as an application.