On Strong Stability and Stabilizability of Linear Systems of Neutral Type

We consider a differential system of neutral type with distributed delay.We obtain a precise norm estimation of semigroup generated by the operator corresponding to the system in question. Our result is based on a spectral analysis of the operator and some uniform estimation of norms of the exponentials of matrices. We also discuss the stability properties of corresponding solutions and the existence of the fastest growing solution.

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“…O'Connor, T.J. Tarn [11], R. Rabah, G.M. Sklyar [12,13], R. Rabah et al [16], S.M. Verduyn Lunel, D.V.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Growth of Solutions of Neutral Type Systems

Sklyar

Polak

2013

Appl Math Optim

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“…We also note that regular stabilizability for a particular case of the control systems (1.7) had been considered in [24,25]. Before giving the proof, let us discuss the conditions (1) and (2).…”

Section: Stabilizability By Regular Feedbackmentioning

confidence: 99%

Stability and stabilizability of mixed retarded-neutral type systems

2011

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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.Mathematics Subject Classification. 34K40, 34K20, 93C23, 93D15.

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“…For linear neutral type systems and hyperbolic partial differential equations they are different: such systems may be asymptotically stable but not exponentially stable and then the same situation occurs for stabilizability (see for example [1,18,23,25]). This situation is related to the location of the spectrum near the imaginary axis (see our paper about the stability problem [17]). For the neutral type systems, as for other infinite dimensional systems (for exemple hyperbolic partial differential systems), the spectrum may contain an infinite set close to the imaginary axis.…”

Section: Introductionmentioning

confidence: 96%

“…For the neutral type systems, as for other infinite dimensional systems (for exemple hyperbolic partial differential systems), the spectrum may contain an infinite set close to the imaginary axis. In [17,18] we gave an analysis of this situation on stability conditions. It is shown that even complete information on the location of the spectrum of the operator A (and A + BF ) does not provide the description of the cases when the system is stable or unstable.…”

Section: Introductionmentioning

confidence: 99%

On Pole Assignment and Stabilizability of Neutral Type Systems

Rabah

Sklyar

Rezounenko

2009

Topics in Time Delay Systems

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Summary. In this note we present a systematic approach to the stabilizability problem of linear infinite-dimensional dynamical systems whose infinitesimal generator has an infinite number of instable eigenvalues. We are interested in strong non-exponential stabilizability by a linear feed-back control. The study is based on our recent results on the Riesz basis property and a careful selection of the control laws which preserve this property. The investigation may be applied to wave equations and neutral type delay equations.

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On Strong Stability and Stabilizability of Linear Systems of Neutral Type

Cited by 18 publications

References 16 publications

Asymptotic Growth of Solutions of Neutral Type Systems

Asymptotic Growth of Solutions of Neutral Type Systems

Stability and stabilizability of mixed retarded-neutral type systems

On Pole Assignment and Stabilizability of Neutral Type Systems

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