Mathematical Analysis of Optimal Control Problems Under Uncertainty

Abstract

This chapter is focussed on existence theory for the solutions of robust and risk averse optimal control problems. As a first step, the classical variational theory of deterministic PDEs is extended to the case of random PDEs. This variational theory may be developed by using either the formalism of tensor product of Hilbert spaces or abstract functions, i.e., functions with values in Banach or Hilbert spaces. However, tensor products of Hilbert spaces have the advantage that the numerical approximation of such random PDEs becomes very natural in such a formalism.