JOURNAL OF MATI-I~MATICAL ANALYSISAND APPLICATIONS 116, 378--414 (1986) Riccati Equations for Nonsymmetric and Nondissipative Hyperbolic Systems with L2-Boundary Controls* S. CHANG AND I. LASIECKA Mathematics Department, University of Florida, Gainesville, Florida 32611 Submitted by A. Schumitzky INTRODUCTION We consider a differential operator of the form A(x, ~)y=_ ~ A:(x)~jy+B(x)y j=l where y(x) is a k-vector and ~j = ~/Oxj. The coefficients Aj, B are smooth k x k matrix-valued functions defined on an open bounded domain Q ~ R m with boundary F. We assume the following: H.1. A(x, 6) is strictly hyperbolic; i.e., ~'=l Aj(x)¢j has k distinct real eigenvalues for all ~ e Rm\ {0 } and x e Q. H.2. The boundary F is noncharacteristic; i_e., detAN(x)¢O 1 for x • F, where AS(x)=-Y~=l AiNj(x); ~= (NI,..., N,,) the inward unit nor- mal. Boundary conditions are imposed with the aid of a boundary operator M(x) which is a smooth l x k matrix-valued function, where l stands for the number of negative eigenvalues of A N. We assume the following: H.3. rank M(x) = l, x ~ F. H.4. (Kreiss condition) The frozen (at the boundary point) mixed problem has no eigenvalues or generalized eigenvalues with nonnegative real parts. * This research was supported in part by the National Science Foundation under Grant DMS-8301668 and by Air Force Office Scientific Research under Grant AFOSE84-0365. 1 From H 1 and H2, by smooth change of coordinates we may assume A, = [o ~7~ o+], where A~ = diag(a 1,..., at) < 0 and A + = diag(at+ 1 ..... ak) > 0. 378 0022-247X/86 $3.00 Copyright © 1986 by AcademicPress, Inc. All rights of reproductionin any form reserved.