\({\mathcal H}_\infty\) control of nonlinear systems: Differential games and viscosity solutions. (English) Zbl 0926.93019

Soravia, Pierpaolo

SIAM J. Control Optimization 34, No. 3, 1071-1097 (1996).

This paper considers a general, nonlinear, controlled, dynamical system subject to unknown disturbances \(\dot y= f(y,a,b)\), with output or response \(h(y,a,b)\), where \(a\) is the control and \(b\) is the disturbance. The disturbances are modelled deterministically as functions of time, and one has to optimize the performance of the system using the worst case criterion. \({\mathcal H}_\infty\) optimal control theory is a deterministic way to tackle the problem. The theory of differential games and the study of the associated Hamilton-Jacobi-Isaacs equation appear to be basic tools of the theory. The author proves that the existence of a continuous, local viscosity supersolution of Isaacs equation corresponding to the \({\mathcal H}_\infty\) control problem is sufficient for its solvability. It is also shown that the existence of a lower-semicontinuous viscosity supersolution is necessary.

Reviewer: S.Shih (Tianjin)

MSC:

93B36	\(H^\infty\)-control
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
91A23	Differential games (aspects of game theory)

Keywords:

nonlinear system; \({\mathcal H}_\infty\) optimal control; differential games; Hamilton-Jacobi-Isaacs equation; viscosity supersolution