Robust feedback switching control: dynamic programming and viscosity solutions

TL;DR

This paper develops a novel robust feedback switching control framework using dynamic programming and viscosity solutions, addressing model misspecification and uncertainty in a two-player game setting.

Contribution

It introduces feedback (closed-loop) switching strategies and applies the stochastic Perron method to characterize the value function as a viscosity solution.

Findings

Proves the value function is the unique viscosity solution of the HJB system.

Establishes the dynamic programming principle for robust feedback switching control.

Develops the stochastic Perron method in a new robust control context.

Abstract

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 method in this framework, and prove that it produces a viscosity sub and supersolution to a system of Hamilton-Jacobi-Bellman (HJB) variational inequalities, which envelope 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 byproduct the dynamic programming principle for robust feedback switching control problem.