Interior penalty approximation for optimal control problems. Optimality conditions in stochastic optimal control theory.

Summary Abstract: This thesis is divided into two parts. In the first part, we focus on deterministic optimal control problems and study interior approximations for two model problems with non-negativity constraints on the control. The first model is an optimal control problem with a quadratic cost function and dynamics governed by an ordinary differential equation. For a general class of interior penalty functions, we show how to compute the principal term of the pointwise state and adjoint state expansion. Our main argument is based on the following fact: if the optimal control for the initial problem satisfies the strict complementarity conditions for the Hamiltonian except at a finite number of times, the estimates for the penalized optimal control problem can be obtained from the estimates for an associated stationary problem. Our results provide several types of approximation quality measures for the penalization technique: error estimates for the control, error estimates for the state and adjoint state and also error estimates for the value function. The second model is the optimal control problem of a semi-linear elliptic equation with homogeneous Dirichlet conditions at the edge, the control being distributed on the domain and positive. The approach is the same as for the first model, i.e. we consider a family of penalized problems, whose solution defines a central trajectory that converges to the solution of the initial problem. In this way, we can extend the results, obtained in the framework of differential equations, to the optimal control of semi-linear elliptic equations. In the second part we focus on stochastic optimal control problems. First, we consider a linear quadratic stochastic problem with non-negativity constraints on the control and we extend the error estimates for the logarithmic penalty approximation. The proof relies on the stochastic Pontriaguine principle and a duality argument. Next, we consider a general stochastic control problem with convex constraints on the control. The so-called variational approach allows us to obtain a first and second order development for the state and the cost function, around a local minimum. With these developments we can show general first order optimality conditions and, under a geometric assumption on the set of constraints, second order necessary conditions are also established.