Stability and stabilizability of mixed retarded-neutral type systems

TL;DR

This paper investigates the stability and stabilizability of mixed retarded-neutral type systems with singular neutral terms, focusing on spectral properties and asymptotic behavior when exponential stability is not achievable.

Contribution

It introduces combined techniques using Riesz bases and resolvent boundedness to analyze asymptotic stability and stabilizability of complex mixed retarded-neutral systems.

Findings

Conditions for asymptotic non-exponential stability are established.

A new approach to analyze stabilizability of unstable systems is proposed.

Extends previous stability and stabilizability results for such systems.

Abstract

We analyze the stability and stabilizability properties of mixed retarded-neutral type systems when the neutral term is allowed to be singular. Considering an operator model of the system in a Hilbert space we are interesting in the critical case when there exists a sequence of eigenvalues with real parts approaching to zero. In this case the exponential stability is not possible and we are studying the strong asymptotic stability property. The behavior of spectra of mixed retarded-neutral type systems does not allow to apply directly neither methods of retarded system nor the approach of neutral type systems for analysis of stability. In this paper two technics are combined to get 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 technics introduced allow to analyze the concept of regular strong stabilizability for mixed retarded-neutral type systems. The present paper extends the results by R. Rabah, G.M. Sklyar, A.V. Rezounenko on stability obtained in [J. Diff. Equat., 214(2005), No. 2, 391-428] and on stabilizability from [J. Diff. Equat., 245(2008), No. 3, 569-593].