Differential Games and Nonlinear \boldmath$\cal H_\infty$ Control in Infinite Dimensions
Maciej Kocan, Pierpaolo Soravia · SIAM Journal on Control and Optimization · 2000
This paper studies the ${\cal H}_\infty$ 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 ${\cal H}_\infty$ 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 ${\cal H}_\infty$ problem has to be relaxed by using the theory of differential games. We apply our results to the one phase Stefan problem.