Numerical Studies of the Linear Quadratic Control Problem for Retarded Systems with Delay in Control

Abstract

In this paper we report on the results of our continuing efforts on using the averaging approximation scheme for retarded functional differential equations. The central focus of this paper is our numerical studies of constructing feedback soLutions to linear quadratic regulator (LQR) problems for retarded systems with delay in control. For completeness, we shall also give a brief summary and discussion of an abstract approximation framework and convergence theory developed previously by Ito and Tran in[9]. In [9]we presented an approximation framework for the numerical treatment of algebraic Riccati equations for a class of linear infinite dimensional systems with unbounded input and output operators studied by Pritchard and Salamon in[19] In this paper we will call it the Pritchard-Salamon clans. This approximation theory which yields convergence of the approximating Riccati operators as well as convergence of the approximating gain operators extends earlier results developed in[7][2][8]in which the input and output operators are assumed to be bounded to the unbounded cases. The main features which distinguish the work in [9]from other work existing in the literature, see e.g.[11][14][15] are the assumptions on the smoothness of the underlined semigroup and the observation map. Because of the smoothness assumptions, the algebraic Riccati soLution has a smoothing property which in turn implies boundedness of the feedback gain operator. Although the theory developed in[9]does not cover many important boundary control problems studied by Lasiecka and Triggiani in [13] and Flandoli, Lasiecka, and Triggiani in[6] for example, it does enable us to treat the control problem governed by delay differential equations with delays in control and observation.