On structured perturbations for two classes of linear infinite-dimensional systems
The authors investigate two classes of infinite-dimensional systems described by an abstract differential equation \(\dot x=(A+B\Delta C)x\) on a Hilbert space, where \(A\), \(B\) and \(C\) are linear, possible unbounded operators and \(\Delta\) is an unknown, linear, bounded perturbation. The two classes are defined in terms of the properties imposed on the triple \((A,B,C)\). It is shown that for every \(\Delta\) the perturbed system given by \((A+E\Delta F,B,C)\) inherits all the properties of the unperturbed system \((A,B,C)\) if \((A,E,F)\) and \((A,B,C)\) are in the same class. The results in this paper can be considered as generalizations of results of \textit{R. F. Curtain}, \textit{H. Logemann}, \textit{S. Townley}, and \textit{H. Zwart} [J. Math. Syst. Estim. Control 4, 493-496 (1994; Zbl 0815.93046)]. However, the theory is less general than the perturbation theory developed for regular systems by \textit{G. Weiss} [Math. Control Signals Syst. 7, 23-57 (1994; Zbl 0819.93034)]. Since the classes of systems considered in this paper are strictly smaller than the class of regular systems, the invariance results in this paper are more specific than those by Weiss [loc. cit.].
scientific article; zbMATH DE number 911965
scientific article