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Linear Volterra-Stieltjes integral equations and control. (English) Zbl 0537.49021

Equadiff 82, Proc. int. Conf., Würzburg 1982, Lect. Notes Math. 1017, 67-72 (1983).
[For the entire collection see Zbl 0511.00014.]
The purpose of this paper is to study some aspects of controllability concerning linear Volterra-Stieltjes integral equations. Volterra- Stieltjes integral equations are considered in many works:
Here we work in the context of C. S. Hönig [”Volterra Stieltjes- integral equations” (1975; Zbl 0307.45002)]. The development of the control theory for this type of equation has an intrinsic interest, since it encloses very general classes of evolutive systems. It comprises, for instance, the linear Stieltjes integral equations, \[ (L)\quad y(t)- x+\int^{t}_{a}dA(s)\cdot y(s)=g(t)-g(a),\quad(a\leq t\leq b), \] the linear delay differential equations and Volterra integral equations [see ibid., pp. 81-94]. Moreover, it is easy to construct very simple models of a perturbated control system, where the process that describes the transfering of the optimal instantaneous controls (with respect to the perturbation) is of linear Volterra-Stieltjes type (see for this direction the book of R. K. Miller [”Nonlinear Volterra integral equations” (1971; Zbl 0448.45004)], p. 67.

MSC:

93B03 Attainable sets, reachability
49K99 Optimality conditions
45D05 Volterra integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)