H-infinity control of nonlinear systems: Differential games and viscosity solutions

Dealing with disturbances is one of the most important questions for controlled systems. H∞ optimal control theory is a deterministic way to tackle the problem in the presence of unfavorable disturbances. The theory of differential games and the study of the associated Hamilton-Jacobi-Isaacs equation appear to be basic tools of the theory. We consider a general, nonlinear system and prove that the existence of a continuous, local viscosity supersolution of the Isaacs equation corresponding to the H∞ control problem is sufficient for its solvability. We also show that the existence of a lower semicontinuous viscosity supersolution is necessary.