Stochastic approximation methods for PDE constrained optimal control problems with uncertain parameters

This thesis work focuses on optimal control of partial differential equations (PDEs) with uncertain parameters, treated as a random variables. In particular, we assume that the random parameters are not observable and look for a deterministic control which is robust with respect to the randomness. The theoretical framework is based on adjoint calculus to compute the gradient of the objective functional. Unlike the deterministic case, we have a set of PDEs indexed by some uncertain parameters and the objective functional we consider includes some risk measure (e.g. expectation, variance, quantile (Value at Risk), ...) to take care of all realizations of our uncertain parameters. Introducing the regular class of coherent risk measure, results of convergence and regularity of the optimal control have been derived.