We study a class of infinite horizon optimal control problems with a state delay, and investigate the dynamic programming approach which leverages the sufficient optimality conditions and provides a closed-loop solution. Importantly, the well-known Lyapunov–Krasovskii functional is applied to relate the solution of the problem to the solution of a set of three Riccati-type matrix differential equations. We then present an analytic-based approach to solve the resultant equations and subsequently provide a suboptimal closed-loop solution for the considered problem. We prove the uniform convergence of the proposed approach and show that the presented closed-loop system is asymptotically stable in the Lyapunov sense. Furthermore, the observability of the linear time-delay system is discussed and proved. Finally, numerical examples illustrate the efficiency of the proposed method.

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