The Linear-Quadratic Control Problem for Retarded Systems with Delays in Control and Observation

THE OBJECT of this paper is to solve the linear-quadratic control problem (LQCP) for retarded functional differential equations (RFDE) with delays in the input and output variables. This will be done within the general semigroup-theoretic framework which has been developed in [28]. For retarded systems with undelayed input and output variables the LQCP has been studied by various authors for about twenty years. We mention the work of Krasovskii [20], Kushner & Barnea [21], Alekal, Brunovskii, Chyung & Lee [1], Delfour & Mitter [14], Curtain [6], Manitius [25], Delfour, McCalla & Mitter [13], Delfour, Lee & Manitius [11], Delfour [8], Banks & Burns [2]. First results on systems with a single-point delay in the state and control variables can be found in Koivo & Lee [19] and Kwong [22]. Ichikawa [16] has developed a comprehensive evolution equation approach for the treatment of the LQCP for RFDEs with input delays. His idea was to include a past segment of the input function in the state of the system. A completely different approach to this problem has been developed by Vinter & Kwong [30] for RFDEs with distributed input delays. Their approach has been generalized to RFDEs with general delay, in the state and control variables by Delfour [9, 10] and to neutral systems by Karrakchu [18] in her recent thesis. The LQCP for neutral systems with output delays has been studied by Datko [7] and Ito & Tarn [17].