Construction of Lyapunov Functions for Piecewise-Deterministic Markov Processes

The purpose of this contribution is twofold: 1) to present for the first time a Lyapunov function that proves exponential ergodicity of a process studied by the authors in [1], where the problem of controlling the probability density of a swarm of robotic agents was solved; 2) to introduce alongside the method used to construct this Lyapunov function, which is of interest in its own since it may be applicable to a wide class of piecewise-deterministc Markov processes. Our method searches for the Lyapunov function that maximizes a measure of the rate of convergence that appears in the theory of large deviations. Analytical solutions are often possible as shown by examples.