Optimization of stochastic uncertain systems: large deviations and robustness for partially observable diffusions

This paper is concerned with stochastic control systems, in which the pay-off is described by the relative entropy between the nominal measure and the uncertain measure, while the uncertain measures satisfy certain energy inequality constraints. With respect to this formulation two problems are defined. The first, seeks to minimize the relative entropy over the set of unknown measures, which satisfy inequality constraints. The second seeks to maximize over the set of admissible control laws, the minimum value of relative entropy induced by the uncertain measures among those, which satisfy inequality constraints. The second problem is equivalent to a minimax problem, while the first is an optimization problem with respect to a fix control law. Certain monotonicity properties of the optimal solution are discussed, while relations to the well-known Cramer´s theorem of large deviations are introduced. In addition, connections to minimax games of partially observable stochastic systems and to risk-sensitive control problems are investigated.