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Stability and stabilizability of mixed retarded-neutral typesystems∗

Published online by Cambridge University Press:  19 September 2011

Rabah Rabah ,
Grigory Mikhailovitch Sklyar  and
Pavel Yurevitch Barkhayev
Rabah Rabah
Affiliation:
IRCCyN/École des Mines de Nantes, 4 rue Alfred Kastler, BP 20722, 44307 Nantes Cedex 3, France. rabah@emn.fr
Grigory Mikhailovitch Sklyar
Affiliation:
Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland; sklar@univ.szczecin.pl
Pavel Yurevitch Barkhayev
Affiliation:
Dept. of Diff. Equat. and Control, Kharkov National University, 4 Svobody sqr., 61077 Kharkov, Ukraine; pbarhaev@inbox.ru
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Abstract

We analyze the stability and stabilizability properties of mixed retarded-neutral typesystems when the neutral term may be singular. We consider an operator differentialequation model of the system in a Hilbert space, and we are interested in the criticalcase when there is a sequence of eigenvalues with real parts converging to zero. In thiscase, the system cannot be exponentially stable, and we study conditions under which itwill be strongly stable. The behavior of spectra of mixed retarded-neutral type systemsprevents the direct application of retarded system methods and the approach of pureneutral type systems for the analysis of stability. In this paper, two techniques arecombined to obtain the conditions of asymptotic non-exponential stability: the existenceof a Riesz basis of invariant finite-dimensional subspaces and the boundedness of theresolvent in some subspaces of a special decomposition of the state space. For unstablesystems, the techniques introduced enable the concept of regular strong stabilizabilityfor mixed retarded-neutral type systems to be analyzed.

Keywords

Retarded-neutral type systemsasymptotic non-exponential stabilitystabilizabilityinfinite dimensional systems
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 656 - 692
Copyright
© EDP Sciences, SMAI, 2011

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