**Posted Dec 1, 1995**

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

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

Download postscipt file (18 pages)
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.