Semistability for time-varying discontinuous dynamical systems with application to agreement problems in switching networks

Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. In this paper, we extend the theory of semistability to discontinuous time-varying dynamical systems. In particular, Lyapunov-based tests for semistability, weak semistability, as well as uniform semistability for nonautonomous differential inclusions are established. Using these results we develop a framework for designing semistable protocols in switching dynamical networks with time-dependent and state-dependent communication topologies.