Stochastic optimal control subject to variational norm uncertainty: Viscosity subsolution for generalized HJB inequality

Abstract

This paper is concerned with optimization of stochastic uncertain systems, when systems are described by measures and the pay-off by a linear functional on the space of measure, on general abstract spaces. Robustness is formulated as a minimax game, in which the control seeks to minimize the pay-off over the admissible controls while the measure aims at maximizing the pay-off over the total variational distance uncertainty constraint between the uncertain and nominal measures. This paper is a continuation of the abstract results in [1], where existence of the maximizing measure over the total variational distance constraint is established, while the maximizing pay-off is shown to be equivalent to an optimization of a pay-off which is a linear combination of L1 and L∞ norms. The maximizing measure is constructed from a convex combination of a sequence of tilted measures and the nominal measure. Here emphasis is geared towards the application of the abstract results to uncertain continuous-time controlled stochastic differential equations, in which the control seeks to minimize the pay-off while the measure seeks to maximize it over the total variational distance constraint. The maximization over the total variational distance constraint is resolved resulting in an equivalent pay-off which is a non-linear functional of the nominal measure of non-standard form. The minimization over the admissible controls of the non-linear functional is addressed by deriving a HJB inequality and viscosity subsolution. Throughout the paper the formulation and conclusions are related to previous work found in the literature. ©2009 IEEE.