Optimal Control of Uncertain Stochastic Systems Subject to Total Variation Distance Uncertainty

This paper is concerned with optimization of uncertain stochastic systems, in which uncertainty is described by a total variation distance constraint between the measures induced by the uncertain systems and the measure induced by the nominal system, while the pay-off is a linear functional of the uncertain measure. Robustness at the abstract setting is formulated as a minimax game, in which the control seeks to minimize the pay-off over the admissible controls while the uncertainty aims at maximizing it over the total variation distance constraint. By invoking the Hanh-Banach theorem, it is shown that the maximizing measure in the total variation distance constraint exists, while the resulting pay-off is a linear combination of L1 and L∞ norms. Further, the maximizing measure is characterized by a linear combination of a tilted measure and the nominal measure, giving rise to a pay-off which is a non-linear functional on the space of measures to be minimized over the admissible controls. The abstract formulation and results are subsequently applied to continuous-time uncertain stochastic controlled systems, in which the control seeks to minimize the pay-off while the uncertainty aims to maximize it over the total variation distance constraint. The minimization over the admissible controls of the non-linear functional pay-off is addressed by developing a generalized principle of optimality or dynamic programming equation satisfied by the value function. Subsequently, it is proved that the value function satisfies a generalized Hamilton-Jacobi-Bellman (HJB) equation. Finally, it is shown that the value function is also a viscosity solution of the generalized HJB equation. Throughout the paper the formulation and conclusions are related to previous work found in the literature.