State space theory of linear time invariant systems with delays in state, control, and observation variables. I, II
These two papers develop a state space theory for an abstract control system which includes a very large class of linear time-invariant hereditary systems: delay-differential equations, integro-differential equations, Volterra-Stieltjes integral equations, functional differential equations of retarded and neutral type, with finite or infinite memory. ``Appropriate states are constructed and associated semigroups and abstract differential equations are obtained. In Part II we emphasize the structural operator approach as in \textit{M. C. Delfour} and \textit{A. Manitius}, ibid. 73, 466-490 (1980; Zbl 0456.34041); and ibid. 74, 359- 381 (1980; Zbl 0464.34050). 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 \textit{R. B. Vinter} and \textit{R. H. Kwong} [SIAM J. Control Optimization 19, 139-153 (1981; Zbl 0465.93043)]. 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 state equations are constructed in such a way that no delay appears in the new observation operator''. The theory developed in these papers presents an alternative to the one contained in \textit{D. Salamon}, Control and observation of neutral systems (1983; Zbl 0546.93041). Most of the results were developed independently. The contents are as follows: Part I: 1) Introduction. 2) Fundamental Results. 3) Transposition, ``transposed'' system and adjoint system. Part II: 4) Structural operators and intertwining theorems. 5) State space theory of systems with control delays. 6) State space theory of systems with delays in control and observation variables.
linear time-invariant hereditary systems
scientific article; zbMATH DE number 4049494
scientific article