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

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, where existence of the maximizing measure over the total variational distance constraint is established, while the maximizing payoff is shown to be equivalent to an optimization of a payoff 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.