Robust feedback switching control: Dynamic programming and viscosity solutions

We consider a robust switching control problem. The controller only observes the evolution of the state process, and thus uses feedback (closed-loop) switching strategies, a non-standard class of switching controls introduced in this paper. The adverse player (nature) chooses open-loop controls that represent the so-called Knightian uncertainty, i.e., misspecifications of the model. The (half) game switcher versus nature is then formulated as a two-step (robust) optimization problem. We develop the stochastic Perron's method in this framework, and prove that it produces a viscosity subsolution and supersolution to a system of HJB variational inequalities, which envelop the value function. Together with a comparison principle, this characterizes the value function of the game as the unique viscosity solution to the HJB equation, and shows as a by-product the dynamic programming principle for the robust feedback switching control problem.