Sensitivity analysis for optimal control problems. Stochastic optimal control with a probability constraint.

Summary This thesis is divided into two parts. In the first part, we study deterministic optimal control problems with constraints and we focus on sensitivity analysis issues. The point of view we adopt is that of abstract optimization. Necessary and sufficient second order optimality conditions play a crucial role and are also studied as such. In this thesis, we are interested in strong solutions. In general, we use this generic term to refer to locally optimal L1-norm controls. By reinforcing the notion of local optimality used, we expect to obtain stronger results. Two tools are used in an essential way: a relaxation technique, which consists in using several controls simultaneously, and a decomposition principle, which is a particular second-order Taylor expansion of the Lagrangian. Chapters 2 and 3 deal with necessary and sufficient second-order optimality conditions for strong solutions of pure, mixed and final state constrained problems. In Chapter 4, we perform a sensitivity analysis for relaxed problems with constraints on the final state. In chapter 5, we perform a sensitivity analysis for a nuclear power generation problem. In the second part, we study stochastic optimal control problems with probability constraints. We study a dynamic programming approach, in which the probability level is seen as an additional state variable. In this framework, we show that the sensitivity of the value function with respect to the probability level is constant along the optimal trajectories. This analysis allows us to develop numerical methods for continuous time problems. These results are presented in Chapter 6, in which we also study an application to asset-liability management.