$\mathcal{H}_\infty $ Control of Nonlinear Systems: Differential Games and Viscosity Solutions

Dealing with disturbances is one of the most important questions for controlled systems. ${\cal H}_\infty$ 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 ${\cal H}_\infty$ control problem is sufficient for its solvability. We also show that the existence of a lower semicontinuous viscosity supersolution is necessary.