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This paper generalizes the results of Vinter and Kwong to a control operator with a more general delay structure than the one in A. Ichikawa and to delay systems with both finite and infinite memories. The linear-quadratic optimal control problem over both finite and infinite time horizons is treated. Several forms of associated abstract Riccati differential and algebraic equations are presented. The properties of the kernel of the solution are studied and a coupled set of matrix ordinary and partial differential equations is obtained.
Michel C. Delfour (1986) studied this question.