Some topics in deterministic and stochastic control : LP methods, PDMP associated to gene networks, controllability
The aim of this synthesis is to present my research activity covering the period elapsed since the terminal year of my PhD program (i.e. the period October 2008 - February 2013). My subjects of research roughly correspond to three main directions, each one presented in a separate section : ∙ Linear programming (LP) methods in deterministic and stochastic control problems; ∙ Control methods in piecewise deterministic Markov processes and their applications to stochastic gene networks; ∙ Controllability properties for linear stochastic systems and related topics. In the first section, we study several classes of deterministic/stochastic control problems with semicontinuous cost. In the stochastic framework, we consider the Mayer problem and optimal stopping for controlled diffusions (corresponding to the paper [G10]), dynamic programming principles (corresponding to the paper [G6]) and control problems with state constraints (corresponding to the paper [G2]). We also study optimal control problems with discounted payoffs in an infinite horizon setting and long-time averaging (corresponding to [G12]), systems driven by stochastic variational inequalities ([G3]), and Zubov's characterization of asymptotic stability domains ([G3]). We investigate the existence of the limit value function for a class of nonlinear stochastic control problems under nonexpansive assumptions and uniform Tauberian theorems (corresponding to [G19]). In the deterministic setting, we consider the linearization and the dynamic programming principles for L^{∞}-control problems (corresponding to the paper [G9]) and for systems with state constraints (in [G1]). We propose a linearization method for min-max control problems (corresponding to [G18]). The common point between these articles is the method employed relying on linearized formulation and viscosity tools. We also present some viability results for singularly perturbed control systems (corresponding to [G13]). The second section is devoted to some contributions to the theory of controlled piecewise deterministic Markov processes (PDMP). We investigate geometric conditions for viability and invariance with respect to controlled PDMPs (corresponding to [G5]). We also provide linear formulations for control problems in this framework (corresponding to [G8] and [G4]). This allows to infer some reachability conditions (cf. [G5]) and to characterize asymptotic stability domains by generalizing Zubov's method (in [G4]). The theoretical results are applied to a class of systems connected to stochastic gene networks (On/Off models, Cook's model for haploinsufficiency and Hasty's model of bistability in bacteriophage λ). The last section presents the study of different types of controllability in connection to: linear jump-diffusions (corresponding to [G7]) and linear control systems of mean-field type (corresponding to [G20]). The arguments involve some viability criteria. A first step towards the study of controllability properties in a Hilbert setting is done by the paper [G11]. In this article, we propose a quasi-tangency approach to stochastic (near)viability with respect to semilinear systems in an infinite-dimensional framework. We have tried to keep the manuscript as self-contained as possible. In order to insure better readability, we have also tried to make the sections independent. However, to keep the manuscript's proportions, we have chosen to limit the redundancy. This is the reason why control problems with state constraints are only presented in the stochastic framework. Also, details for Zubov's method are solely given in the PDMP framework and we only mention the contribution for Brownian diffusions.