Characterization of eventual conditional invariance for hybrid dynamical systems

Abstract

Sufficient conditions to guarantee eventual conditional invariance (ECI) for hybrid dynamical systems are formulated. Roughly speaking, eventual conditional invariance encodes the property of solutions that, when they start from a given set of initial conditions, they reach a target set after a finite amount of time and remain in it for future time. The sufficient conditions for ECI presented in this paper are infinitesimal, in the sense that they need to be checked at points in the state space of the hybrid dynamical system under study, they involve its data and an appropriate choice of Lyapunov-like functions, and rely on solution properties of a continuous-time and a discrete-time scalar system. Rather than assuring finite-time attractivity plus forward invariance of the target set, as found in previous works, the proposed conditions allow the solutions to enter and leave the target set provided that they eventually settle in it after some time. Examples throughout the paper illustrate key concepts and results.