State space theory of linear time invariant systems with delays in state, control, and observation variables, I

Abstract

Part I of this paper studies a coupled mathematical system which provides a good model for important families of linear time-invariant hereditary systems: delay-differential equations, integro-differential equations, Volterra-Stieltjes integral equations, functional differential equations of retarded and neutral types, etc. Appropriate states are constructed and associated semigroups and abstract differential equations are obtained. In Part II we emphasize the structural operator approach as in Delfour and Manitius. Control operators are added to the coupled mathematical system allowing delays in the control variables. Again structural operators are introduced to define the state and obtain abstract differential equations without delays in the control variable as in the work of Vinter and Kwong. Finally observation operators are added which allow for delays in the observation variable and delayed control variables in the observation equation. Again a new state and a state equations are constructed in such a way that no delay appears in the new observation operator thus generalizing the construction of A. Pritchard and D. Salamon.