2406 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000 Finite Horizon State-Feedback Control of Continuous-Time Systems with State Delays Emilia Fridman and Uri Shaked Abstract—The finite horizon control of time-invariant linear sys- tems with a finite number of point and distributed time delays is consid- ered. The controller is obtained by solving coupled Riccati-type partial dif- ferential equations. The solutions to these equations and the resulting con- trollers are approximated by series expansions in powers of the largest delay. Unlike the infinite horizon case, these approximations possess both regular and boundary layer terms. The performance of the closed-loop system under the memoryless zero-approximation controller is analyzed. Index Terms—Asymptotic approximations, –state-feedback control, Riccati type partial differential equations, singular perturbations, time- delay systems. I. PROBLEM FORMULATION Throughout this paper we denote by the Euclidean norm of a vector or the appropriate norm of a matrix. Given , let be the space of the square integrable functions with the norm and let be the space of the continuous functions on with the norm . We denote . Prime denotes the transpose of a matrix and col denotes a column vector with components and . Consider the system (1) where is the state vector, is the control signal, is the exogenous disturbance, is the observation vector, and , and are constant matrices of appropriate dimen- sions. The -valued function which carries -valued functions on into is defined as follows: (2) where are constant matrices and is a square integrable matrix function. Denote (3) and where is the indicator function for the set , i.e., if and otherwise. Given , and assuming that and , we consider the following performance index: (4) where (5) Manuscript received July 26, 1999; revised December 23, 1999. Recom- mended by Associate Editor, Y. Yamamoto. The authors are with the Department of Electrical Engineering-Sys- tems, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel (e-mail: emilia@eng.tau.ac.il). Publisher Item Identifier S 0018-9286(00)10012-1. and where are matrices denoting the initial weightning condition. The problem is to find a state-feedback con- troller which ensures that for all and for all initial conditions . In the infinite horizon case such a controller (for zero initial condition ) has been designed in [1]–[6] (see also references therein). In [1] and [2] the controller has been obtained by solving Riccati operator equations. In [3] and [4] delay-independent and in [6] delay-dependent memoryless controllers have been designed. In [5] the controller (with memory) has been derived from Riccati-type partial differential equa- tions (RPDEs) or inequalities, and the solution of the RPDEs has been approximated by expansions in the powers of the delay. In [7] the gra- dient of with respect to at has been computed. In [8] and [9] bounded real criteria have been obtained. Asymptotic series solutions of systems with small delay have been constructed in [10]–[12]. In many engineering cases (target maneuver, missile guidance, etc.) a control session of limited time length is needed. In such cases the ef- fect of the initial conditions is most important and the results of [1]–[6] cannot provide a satisfactory control strategy. In the present paper, we generalize the results of [5] to the finite horizon case. Unlike [5], the required controllers are time-varying, they are obtained by solving cou- pled finite horizon RPDEs. For small delays, similarly to the case of singularly perturbed systems (see [10]–[13]), the controllers are af- fected by the boundary-layer phenomenon. The main contribution of the paper is the construction, for the first time, of an asymptotic solu- tion to the important class of finite horizon RPDEs that are encountered with the finite-horizon LQ control (see [14]), and with the control. Proofs of Theorem 1 and Lemma 2 are given in the Appendix. II. MAIN RESULTS A. -Controller Design Consider the following RPDEs with respect to the -matrices , and : (6) (7) (8) (9) (10) where . A solution of (6)–(10) is a triple of -matrices , 0018–9286/00$10.00 © 2000 IEEE