Foundations for approximation based analysis of stability properties of hybrid systems

We discuss pre-orders for reasoning about stability properties of hybrid systems, including Lyapunov stability, asymptotic stability, input-to-state stability and incremental input-to-state stability. We strengthen the classical notion of equivalence between processes, namely, bisimulation with continuity and uniformity conditions and show that the new notions preserve the stability properties for hybrid systems. We demonstrate the usefulness of our results by casting various classical methods for proving stability as constructing simpler systems which are related to the original systems by uniformly continuous notions of pre-orders and proving the simpler systems to be stable.