Optimization of nonlinear stochastic uncertain relaxed controlled systems: entropy rate functionals and robustness

This paper is concerned with nonlinear stochastic uncertain relaxed controlled diffusions, in which the pay-off is described by the relative entropy between the nominal measure and the uncertain measure, when the uncertain measure satisfies 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 relaxed 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, the implication of these results to minimax games for fully observable stochastic systems in which the strategies are measures are delineated and relations to risk-sensitive control problems are investigated.