ABSTRACT
In the previous chapter we introduced the notion of hybrid dynamical system (HDS) defined on generalized time T and we showed that the qualitative properties of an HDS defined on generalized time can always be reduced to studying a corresponding hybrid dynamical system defined on real time R+ having motions that may not necessarily be continuous on J?+, or that may be continuous but not necessarily continuously defferentiable on R+ (see Proposition 8.4.11). In Chapters 2 through 7, our focus (for continuous time systems) was primarily on dynamical systems with motions that are continous with respect to real time. Accordingly, if we are to be able to study the qualitative properties of the different kinds of HDS discussed in Chapter 8, we need to develop a qualitative theory for discontinous dynamical systems (DDS) denned on R+. The results of such a theory should also hold for dynamical systems defined on R+ with motions that are continuous on R+ but may or may not be continuously differentiable with respect to time. This will be accomplished in the present chapter. In doing so, we will concentrate on general results for DDS (resp., HDS) that are in the spirit of corresponding results for systems with continous motions developed in Chapters 2 through 7. We will apply the results of the present
chapter in the analysis of several important specific classes of HDS (resp., DDS) in the next chapter. There are of course also important classes of HDS that cannot be treated by the general results developed herein.
