Toward a general stability theory for hybrid dynamical systems

Summary form only given. A successful stability theory for general hybrid dynamical systems has to address carefully such fundamental issues as generalized time, the appropriate mathematical setting, and the appropriate models for hybrid dynamical systems. Such systems will usually be discontinuous, they may be finite or infinite dimensional, and they may be generated by equations or inequalities (e.g., ordinary differential equations, ordinary differential inequalities, ordinary difference equations (or ordinary difference inequalities), functional differential equations, partial differential equations, Volterra integrodifferential equations, and the like) or they may involve an “equation free” characterization (e.g., discrete event systems, Petri nets, temporal logic elements, and so forth), or they may be determined by appropriate combinations of the above. To be reasonably complete, such a stability theory should include the usual Lyapunov and Lagrange stability results, converse theorems, and a comparison theory. We focus primarily on results which comprise a comparison theory for hybrid dynamical systems. In their most general form, these results are phrased in terms of stability preserving mappings, while more specific cases involve vector Lyapunov functions. The latter, in turn, can be specialized to yield the classical Lyapunov stability results. Furthermore, the results involving vector Lyapunov functions may be applied in a natural manner in the stability analysis of interconnected hybrid dynamical systems