Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov's Method

Dan Goreac (1) , Oana Silvia Serea (2)

1 PS
2 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations

We aim at characterizing domains of attraction for controlled piecewise deterministic processes using an occupational measure formulation and Zubov's approach. Firstly, we provide linear programming (primal and dual) formulations of discounted, infinite horizon control problems for PDMPs. These formulations involve an infinite-dimensional set of probability measures and are obtained using viscosity solutions theory. Secondly, these tools allow to construct stabilizing measures and to avoid the assumption of stability under concatenation for controls. The domain of controllability is then characterized as some level set of a convenient solution of the associated Hamilton-Jacobi integral-differential equation. The theoretical results are applied to PDMPs associated to stochastic gene networks. Explicit computations are given for Cook's model for gene expression.

Mots clés: Stability domain; Viscosity solutions; PDMP; Occupational measure; Zubov's method; Gene networks

Domaines: Optimisation et contrôle [math.OC]; Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]

Dan Goreac, Oana Silvia Serea. Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov's Method. Applied Mathematics and Optimization, 2012, 66 (2), pp.209-238. ⟨10.1007/s00245-012-9169-x⟩. ⟨hal-00727728⟩