Posted Dec 1, 1995

Viscosity Solutions of Hamilton-Jacobi Equations Arising In Nonlinear H Control

by J. A. Ball and J. W. Helton

The nonlinear H control problem (i.e. the problem of selecting a stabilizing state or measurement feedback subject to an L2-gain constraint for the closed loop input-output map) recently has been intensively studied (see [BB, BHW, JB, IA1, IA2, vS1, vS2, vS3]). All these works involve the investigation of a Hamilton-Jacobi type equation which is satisfied by the storage or energy function associated with the closed loop system if this function happensto be smooth. However in general this storage function need not be smooth and hence does not correspond to a solution of the Hamilton-Jacobi equation in the classical sense. M. Crandall and P.L. Lions [CL] have discovered a notion of weak, or so-called "viscosity" solution (or "subsolution") for a Hamilton-Jacobi equation which has since been applied in various optimal control and differential game contexts (see e.g. [L, ES]), including applications to systemsgoverned by partial differential equations. The purpose of this paper is to extend one of the existing studies on the nonlinear H control problem (specifically the necessity analysis of the authors in [BHW] on the measurement feedback problem) to the case of a nonsmooth storage function. In this paper we deal only with the L2-gain condition in the formulation of the H control problem; the internal stability side condition is a separate issue which under appropriate conditions can be handled in the same way as in the smooth case (see [IA1, IA2]). In the full-length paper we analyze the state feedback case in detail; here we proceed directly to the measurement feedback problem.