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__One Approach to Numerical Solution of a Finite-Horizon Linear-Quadratic Optimal Control Problem for Time Delay Systems__

V. Glizer and V. Turetsky

**Abstract**

The well known approach to solution of a finite-horizon Linear-Quadratic Control Problem (LQCP) with state delays in dynamics reduces this problem to a set of three Riccati-type matrix functional-differential equations: one ordinary and two partial first-order with two and three independent variables [1]. This set does not allow, in general, an exact solution.
Several approaches to approximate solution of the LQCP with state delays are known: 1) an approximation of the original problem [1]; 2) an asymptotic solution [3, 4]; 3) a finite-difference approximation of the set of Riccati-type equations [2].
In [4], an eliminating the unknown matrix with three arguments from the set of Riccati-type equations, following by a finite-difference approximation of the reduced set, is suggested for a simple example. In this talk, we extend this approach to a much more general case. This approach reduces considerably a computer memory and a computing time, which is illustrated by examples.

**Bibliography**

[1]
M.C. Delfour, The linear quadratic optimal control problem for hereditary differential systems: theory and numerical solution, Appl. Math. Optim. 3 (1977), 101-162.

[2]
D.H. Eller, J.K. Aggarwal and H.T. Banks, Optimal control of linear time-delay systems, IEEE TAC 14 (1969), 678-687.

[3]
V.Y. Glizer, Linear-quadratic optimal control problem for singularly perturbed systems with small delays, Contem. Math., AMS 514 (2010), 155-188.

[4]
V.Y. Glizer, L.M. Fridman and V. Turetsky, Cheap suboptimal control of uncertain systems with state delays: an integral sliding mode approach, TAE954, Technion-IIT, 2005.