Volume 18, Number 2, April-June 2012
|Page(s)||401 - 426|
|Published online||13 April 2011|
Viability, invariance and reachability for controlled piecewise deterministic Markov processes associated to gene networks
UniversitéParis-Est, Laboratoire d’Analyse et Mathématiques
Appliquées, UMR 8050, Boulevard
Descartes, Cité Descartes, 77450
Received: 11 February 2010
Revised: 22 November 2010
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
Key words: Viscosity solutions / PDMP / gene networks
© EDP Sciences, SMAI, 2011
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