Differential games and nonlinear H-infinity control in infinite dimensions

This paper studies the H∞ control problem for a nonlinear, unbounded, infinite dimensional system with state constraints. We characterize the solvability of the problem by means of a Hamilton-Jacobi Isaacs (HJI) equation, proving that the H∞ problem can be solved if and only if the HJI equation has a positive definite viscosity supersolution, vanishing and continuous at the origin. In order to do so, the standard definition of the H∞ problem has to be relaxed by using the theory of differential games. We apply our results to the one phase Stefan problem.