Stability of hybrid systems

We introduce a model of hybrid systems as a combination of discrete state and continuous state systems. The continuous state space is divided into regions so that in every region, depending on the discrete state of hybrid system, the continuous state system dynamic functions which are called representative functions are found. The switching situation in a region is studied and the continuous state dynamics to be used in this situation is defined. It is shown that after a switching mode the system behavior may be uncertain due to the high frequency discrete state switchings. The classical stability of the origin of the continuous state space is defined. Following the Lyapunov theory, some stability theorems are provided.