This paper deals with stability analysis of hybrid systems. Such systems are characterized by a combination of continuous dynamics and logic based switching between discrete modes. Lyapunov theory is a well known methodology for the stability analysis of linear and nonlinear systems in control system literature. Construction of Lyapunov functions for hybrid systems is generally a difficult task, but once these functions are defined, stabilization of the system is straight-forward. The sum of squares (SOS) decomposition and semidefinite programming has also provided an efficient methodology for analysis of nonlinear systems. The computational method used in this paper relies on the SOS decomposition of multivariate polynomials. By using SOS, we construct a (some) Lyapunov function(s) for the hybrid system. The reduction techniques provide numerical solution of large-scale instances; otherwise they will be practically unsolvable. The introduced method can be used for hybrid systems with linear or nonlinear vector fields. Some examples are given to demonstrate the capabilities of the proposed approach.

Multimedia multicast with two servers based on the multiterminal source coding is studied in some previous researches. Due to the possibility of providing an approach for practical code design for more than two correlated sources in IMTSC/CEO setup, in this paper, the framework of Slepian-Wolf coded quantization is extended and a practical code design is presented for IMTSC/CEO with the number of encoders greater than two. Then the multicast system based on the IMTSC/CEO is applied to the cases with three, four and five servers. Since the underlying code design approach for the IMTSC/CEO problem has the capability of applying to an arbitrary number of active encoders, the proposed MMBMSC method can also be used with an arbitrary number of servers easily. Also, explicit expressions of the expected distortion with an arbitrary number of servers in the MMBMSC system are presented. Experimental results with data, image and video signals show the superiority of our proposed method over the conventional solutions and over the MMBMSC system with two servers.