Stability and stabilizability of mixed retarded-neutral type systems^∗

¹ IRCCyN/École des Mines de Nantes, 4 rue Alfred Kastler, BP 20722, 44307 Nantes Cedex 3, France
rabah@emn.fr
² Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
sklar@univ.szczecin.pl
³ Dept. of Diff. Equat. and Control, Kharkov National University, 4 Svobody sqr., 61077 Kharkov, Ukraine
pbarhaev@inbox.ru

Received: 12 November 2009
Revised: 17 December 2010

Abstract

We analyze the stability and stabilizability properties of mixed retarded-neutral type systems when the neutral term may be singular. We consider an operator differential equation model of the system in a Hilbert space, and we are interested in the critical case when there is a sequence of eigenvalues with real parts converging to zero. In this case, the system cannot be exponentially stable, and we study conditions under which it will be strongly stable. The behavior of spectra of mixed retarded-neutral type systems prevents the direct application of retarded system methods and the approach of pure neutral type systems for the analysis of stability. In this paper, two techniques are combined to obtain the conditions of asymptotic non-exponential stability: the existence of a Riesz basis of invariant finite-dimensional subspaces and the boundedness of the resolvent in some subspaces of a special decomposition of the state space. For unstable systems, the techniques introduced enable the concept of regular strong stabilizability for mixed retarded-neutral type systems to be analyzed.