1. Chess Review May 10, 2004 Berkeley, CA A stability criterion for Stochastic Hybrid Systems A. Abate, L. Shi, S. Simic and S.S. Sastry University of California at Berkeley

2. Chess Review, May 8, 2003 2 Problem Statement Dynamics: ODE’s, possibly nonlinear (flows have bounded Lipschitz constant) Underlying Markov Chain Temporal transitions (statistically distributed) Single Equilibrium q shared among all domains Reset maps with bounded Lip constant n Domains Vector fields f i ! flows  i Reset maps R ij Steady-state distr. p=[p 1,..., p n ]

3. Chess Review, May 8, 2003 3 Results In the case of LTI dynamical systems with jumping times t i described by finite-mean r.v. ’s: for all i, i=1,..,n E[t i ]= i. Theorem Define:  i=1,..,n Lip(  i i )  i  i,j=1,..,n Lip(R ij )  i P ij If  then equilibrium q is stable in probability (sufficient condition). –If jumping times happen at constant times, result is valid for general NL systems. Applicative Example in Finance.