# 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

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topRabah, 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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