Partially Observed Differential Games, Infinite-Dimensional Hamilton–Jacobi–Isaacs Equations, and Nonlinear $H_\infty $ Control

This paper presents new results for partially observed nonlinear differential games. Using the concept of information state, we solve this problem in terms of an infinite-dimensional partial differential equation, which turns out to be the Hamilton–Jacobi–Isaacs (HJI) equation for partially observed differential games. We give definitions of smooth and viscosity solutions and prove that the value function is a viscosity solution of the HJI equation. We prove a verification theorem, which implies that the optimal controls are separated in that they depend on the observations through the information state. This constitutes a separation principle for partially observed differential games. We also present some new results concerning the certainty equivalence principle under certain standard assumptions. Our results are applied to a nonlinear output feedback $H_\infty $ robust control problem.