mathematics of computation volume 57, number 196 october 1991, pages 639-662 NUMERICAL APPROXIMATIONS OF ALGEBRAIC RICCATI EQUATIONS FOR ABSTRACT SYSTEMS MODELLEDBY ANALYTIC SEMIGROUPS, AND APPLICATIONS I. LASIECKA AND R. TRIGGIANI Abstract. This paper provides a numerical approximation theory of algebraic Riccati operator equations with unbounded coefficient operators A and B , such as arise in the study of optimal quadratic cost problems over the time interval [0, oo] for the abstract dynamics y = Ay + Bu . Here, A is the generator of a strongly continuous analytic semigroup, and B is an unbounded operator with any degree of unboundedness less than that of A . Convergence results are provided for the Riccati operators, as well as for all the other relevant quantities which enter into the dynamic optimization problem. The present numerical theory is the counterpart of a known continuous theory. Several examples of partial differential equations with boundary/point control, where all the required assumptions are verified, illustrate the theory. They include parabolic equations with L2"Dirichlet control, as well as plate equations with a strong degree of damping and point control. 1. Introduction: continuous and discrete optimal control problems; main results; literature 1.1. Statement of the continuous problem: Assumptions and main results. Con- sider the following optimal control problem: Given the dynamical system, (1.1) yt = Ay + Bu; y(0)=y0£H, minimize the quadratic functional /•oo (1.2) J(u,y)= / [H/WOIIz + NOIIt,]^ Jo over all u £ L2(0, oo ; U), with y a solution of (1.1) with control function u . Received March 12, 1990; revised August 8, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 65K99, 65P05. This research was partially supported by the National Science Foundation under Grant DMS-87- 96320 and by the Air Force Officeof Scientific Research under Grant AFOSR-87-0321. This paper was presented by the first-named author at the Workshop on Computational Aspects of Control held at the Center for the Mathematical Sciences, University of Wisconsin, May 1988. Because of the paper's length, most of its technical proofs are given in the Supplement section of this issue. Should this hinder the reading, the original manuscript—which incorporates the proofs in the body of the paper in a consequential manner—is available by the authors upon request. © 1991 American Mathematical Society 0025-5718/91 $1.00+ $.25 per page 639 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use