Viability, Invariance and Reachability for Controlled Piecewise Deterministic Markov Processes Associated to Gene Networks

We aim at characterizing viability, invariance and some reachability properties of controlled piecewise deterministic Markov processes (PDMPs). Using analytical methods from the theory of viscosity solutions, we establish criteria for viability and invariance in terms of the first order normal cone. We also investigate reachability of arbitrary open sets. The method is based on viscosity techniques and duality for some associated linearized problem. The theoretical results are applied to general On/Off systems, Cook’s model for haploinsufficiency, and a stochastic model for bacteriophage λ. Mathematics Subject Classification. 49L25, 60J25, 93E20, 92C42. Received February 11, 2010. Revised November 22, 2010. Published online April 13, 2011.