Volume 18, Number 3, July-September 2012
|Page(s)||656 - 692|
|Published online||19 September 2011|
Stability and stabilizability of mixed retarded-neutral type systems∗
IRCCyN/École des Mines de Nantes, 4 rue Alfred Kastler, BP 20722, 44307
Nantes Cedex 3,
2 Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
3 Dept. of Diff. Equat. and Control, Kharkov National University, 4 Svobody sqr., 61077 Kharkov, Ukraine
Revised: 17 December 2010
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.
Mathematics Subject Classification: 34K40 / 34K20 / 93C23 / 93D15
Key words: Retarded-neutral type systems / asymptotic non-exponential stability / stabilizability / infinite dimensional systems
© EDP Sciences, SMAI, 2011
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.