Partial Stability and Boundedness of Discontinuous Dynamical Systems

  • A.N. Michel
  • A.P. Molchanov
  • Y. Sun

Abstract

We present results for partial stability of invariant sets and boundedness of motions for discontinuous dynamical systems (DDS) defined on metric space. We first establish a general comparison theory for DDS using stability preserving mappings. Next, we specialize these results by utilizing in particular vector Lyapunov functions as stability preserving mappings. For the scalar case, these results reduce to the Principal Lyapunov Results for partial stability and boundedness of general motions of DDS defined on metric space. We demonstrate the applicability of our results by considering a special class of finite dimensional dynamical systems subjected to impulse effects. We show that for this particular class of DDS, our results are less conservative than existing results. The present results are applicable to a much larger class of DDS than existing results on partial stability and boundedness (including to DDS that cannot be characterized by the usual classical equations and inequalities). Also, in contrast to existing results which pertain primarily to the analysis of equilibria, the present results apply to invariant sets of DDS (including equilibria as a special case). Finally, some of the present results are less conservative than existing ones. The results of the present paper are in the same spirit as our earlier work concerning the partial stability of invariant sets and boundedness of motions, for continuous dynamical systems defined on metric space.
Published
2002-08-01
Section
Articles