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 [1], 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 L-1 and L (infinity) 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.