Multistability analysis of discontinuous dynamical systems via finite trajectory length

Multistability is a global property whereby the solutions of a dynamical system are neither stable nor unstable, but rather alternate between two or more mutually exclusive Lyapunov stable and convergent states over time. In particular, multistable systems give rise to the existence of multiple stable equilibria involving a switching-like behavior between these multiple semistable steady states. This scenario happens in many applications such as spiking neural networks, chemical reaction networks, and agent-based coordination. In this paper, we extend some previous results on the trajectory length based Lyapunov approach to address global stability properties for non-hyperbolic, discontinuous dynamical systems. In particular, the notion of multistability for discontinuous dynamical systems with Carathéory and Filippov solutions is characterized by a series of trajectory-length-based Lyapunov tests, with an interesting feature that they do not depend on a particular equilibrium, do not demand hyperbolicity of equilibria, and do not require the positive definiteness of Lyapunov functions. The results lay a rigorous mathematical foundation for discontinuous control and facilitate such control design.