Stability and stabilizability of mixed retarded-neutral type systems∗

Rabah Rabah; Grigory Mikhailovitch Sklyar; Pavel Yurevitch Barkhayev

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 656-692
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Rabah, Rabah, Sklyar, Grigory Mikhailovitch, and Barkhayev, Pavel Yurevitch. "Stability and stabilizability of mixed retarded-neutral type systems∗." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 656-692. <http://eudml.org/doc/277824>.

@article{Rabah2012,
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. },
author = {Rabah, Rabah, Sklyar, Grigory Mikhailovitch, Barkhayev, Pavel Yurevitch},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Retarded-neutral type systems; asymptotic non-exponential stability; stabilizability; infinite dimensional systems; retarded-neutral type systems},
language = {eng},
month = {11},
number = {3},
pages = {656-692},
publisher = {EDP Sciences},
title = {Stability and stabilizability of mixed retarded-neutral type systems∗},
url = {http://eudml.org/doc/277824},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Rabah, Rabah
AU - Sklyar, Grigory Mikhailovitch
AU - Barkhayev, Pavel Yurevitch
TI - Stability and stabilizability of mixed retarded-neutral type systems∗
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 656
EP - 692
AB - 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.
LA - eng
KW - Retarded-neutral type systems; asymptotic non-exponential stability; stabilizability; infinite dimensional systems; retarded-neutral type systems
UR - http://eudml.org/doc/277824
ER -

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