On the decidability of stability of hybrid systems

Pavithra Prabhakar and Mahesh Viswanathan

A rectangular switched hybrid system with polyhedral invariants and guards, is a hybrid automaton in which every continuous variable is constrained to have rectangular flows in each control mode, all invariants and guards are described by convex polyhedral sets, and the continuous variables are not reset during mode changes. We investigate the problem of checking if a given rectangular switched hybrid system is stable around the equilibrium point 0. We consider both Lyapunov stability and asymptotic stability. We show that checking (both Lyapunov and asymptotic) stability of planar rectangular switched hybrid systems is decidable, where by planar we mean hybrid systems with at most 2 continuous variables. We show that the stability problem is undecidable for systems in 5 dimensions, i.e., with 5 continuous variables.

16th international conference on Hybrid systems: computation and control (HSCC), 2013
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