COMMUNICATIONS IN INFORMATION AND SYSTEMS c  2011 International Press Vol. 11, No. 4, pp. 307-326, 2011 001 STABILITY OF SWITCHED LINEAR SYSTEMS WITH POISSON SWITCHING B. HANLON * , V. TYURYAEV † , AND C. MARTIN ‡ Abstract. We examine the stability of continuous time linear switched systems when the switch- ing times are governed by a Poisson process. We construct an infinite family of polynomial Lyapunov functions that governs the stability of the expected value, as well as higher order moments. The anal- ysis is accomplished by converting the problem to a stochastic process and analyzing this process. Key words: random products, stochastic systems, switching systems, stability 1. Introduction. The study of multimode systems capable of fast switching rates is indeed very challenging. Stochastic control provides a framework for modeling and analysis if some probabilistic data on switching patterns are available. In this paper, we take a different path and ask the simple question: If switching between modes is governed by a Poisson process, how can we carry out a stability analysis of the system? Basic tools to carry out such an analysis are a prerequisite to developing a successful theory of stabilization of switching systems whose switching times are governed by a known probability distribution. From a heuristic viewpoint, stability is a necessary property to accompany solutions to all control paradigms, and, in this sense, we are taking a step to develop new control paradigms. It is our hope that in the future we will be able to answer questions such as stabilization, trajectory planning with closed-loop stability, and tracking. The human muscular control system provides a canonical example of a system with rapid switching. Most of the muscles of the body occur in opposing pairs and only one muscle of the pair can be actively used at any one instant. In a simple movement, such as the movement of the eye in a horizontal plane, the exterior rectus muscle controls the temporal motion and the interior rectus controls the nasal mo- tion. To track an object that is oscillating, using only eye movements, the control switches between the internal and external rectus muscle. See [25] for a more complete analysis of eye movement. In more complicated motion there may be many muscle pairs involved that are switching between opposing muscles and even opposing muscle groups. One only needs to watch a gymnast performing on the pommel horse to see rapid switching between muscle groups. It is reasonably clear in the pommel horse * Department of Statistics, University of Wisconsin, Madison, WI 53706-1510, E-mail: han- lon@stati.wisc.edu † Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409-1024, USA, E-mail: vadim.Tyuryaev@ttu.edu ‡ Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409-1024, USA, E-mail: clyde.f.martin@ttu.edu, the author wishes to acknowledge the hospitality of Kyoto University during the time this research was done. 307