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A multistage reduction technique for feedback stabilizing distributed time-lag systems. (English) Zbl 0629.93046

This paper presents a method which can be used to stabilize a system with state and control delays of the form: \[ \dot x=\int^{0}_{-h}d\alpha (s)x(t+s)+\int^{0}_{-h}d\beta (s)u(t+s). \] A preliminary requirement for this method is that the equation \(A=\int^{0}_{-h}\exp (As)d\alpha (s)\) can be solved with respect to the unknown matrix A. Let \(\Gamma\) be the set of the matrices A which solve this last equation. Then, it is proved that \(\sigma\) (A) is contained in the spectrum of the delay system for each \(A\in \Gamma\). Moreover, a special transformation can be applied to the pairs (x,u) (which solve the delay system) so that the transformed pair is solution of an ordinary differential equation. It is shown, under suitable assumptions, that a feedback which stabilizes this last equation can be used for the task of stabilizing the delay system.
Reviewer: L.Pandolfi

MSC:

93D15 Stabilization of systems by feedback
34K35 Control problems for functional-differential equations
93C05 Linear systems in control theory
93B17 Transformations
93B55 Pole and zero placement problems
93C25 Control/observation systems in abstract spaces
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