Towards a stability theory of general hybrid dynamical systems

In recent work we proposed a general model for hybrid dynamical systems who se states are defined on arbitrary metric space and evolve along some notio n of generalized abstract time. For such systems we introduced the usual co ncepts of Lyapunov and Lagrange stability. We showed that it is always poss ible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties, but defined o n real time R+ = [0, infinity). The motions of this class of systems are in general discontinuous. This class of systems may be finite or infinite dim ensional. For the above discontinuous dynamical systems land hence, for the above hybrid dynamical systems), we established the Principal Lyapunov Sta bility Theorems as well as Lagrange Stability Theorems. For some of these, we also established converse theorems. We demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems. In the present paper we continue the work described above. In doing so, we fi rst develop a general comparison theory for the class of hybrid dynamical s ystems (resp., discontinuous dynamical systems) considered herein, making u se of stability preserving mappings. We then show how these results can be applied to establish some of the Principal Lyaponov Stability Theorems. For the latter, we also state and prove a converse theorem not considered prev iously. Finally, to demonstrate the applicability of our results, we consid er specific examples throughout the paper. (C) 1999 Elsevier Science Ltd. A ll rights reserved.