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Quadratic stability and stabilization of switched dynamic systems with uncommensurate internal point delays. (English) Zbl 1108.93062
Summary: This paper deals with the quadratic stability and linear state-feedback and output-feedback stabilization of switched delayed linear dynamic systems with, in general, a finite number of non-commensurate constant internal point delays. The results are obtained based on Lyapunov’s stability analysis via appropriate Krasovsky-Lyapunov’s functionals and the related stability study is performed to obtain both delay independent and delay dependent results. It is proved that the stabilizing switching rule is arbitrary if all the switched subsystems are quadratically stable and that it exists a (in general, non-unique) stabilizing switching law when the system is polytopic, stable at some interior point of the polytope but with non-necessarily stable parameterizations at the vertices defining the subsystems. It is also proved that two subsystems individually parametrized in different polytopic-type sets which are not quadratically stable might be stabilized by a (in general, non-unique) switching law provided that a convexity-type condition is fulfilled at each existing pair of vertices, one corresponding to each subsystem. Some extensions are provided for a typical class of neutral time-delay systems.

##### MSC:
 93D15 Stabilization of systems by feedback 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93C15 Control/observation systems governed by ordinary differential equations
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