Stability analysis of a general class of hybrid dynamical systems

Hybrid dynamical systems which are capable of exhibiting simultaneously several kinds of dynamic behavior, such as continuous-time dynamics, discrete-time dynamics, jump phenomena, switching and logic commands, discrete events, and the like, are of great current interest. In the present paper we employ a general model of dynamical system suitable in the qualitative analysis of such systems in which generalized time is not represented, as is usually the case, by R+ = [0, infinity) or N = {0, 1, 2, ...}, but by an abstract metric space on which certain suitable hypotheses are imposed. This model of dynamical system allows discontinuous motions, and convergence of motions is relative to generalized time. In the context of the model for hybrid dynamical systems described above we establish the principal Lyapunov stability results for invariant sets and the principal Lagrange stability results for motions. We emphasize that the present work constitutes a continuation of the work initiated by the authors in a previous paper [8]. Some of the results of the present paper are applied in the analysis of a specific class of systems.