H∞-norm and invariant manifolds of systemswith state delays

The problem of nding bounds on the H∞-norm of systems with a nite number of point delays and distributed delay is considered. Su cient conditions for the system to possess an H∞-norm which is less or equal to a prescribed bound are obtained in terms of Riccati partial di erential equations (RPDE’s). We show that the existence of a solution to the RPDE’s is equivalent to the existence of a stable manifold of the associated Hamiltonian system. For small delays the existence of the stable manifold is equivalent to the existence of a stable manifold of the ordinary di erential equations that govern the ow on the slow manifold of the Hamiltonian system. This leads to an algebraic, nite-dimensional, criterion for systems with small delays. c © 1999 Elsevier Science B.V. All rights reserved.