Generalized Riesz basis property in the analysis of neutral type systems

Rabah, Rabah; Sklyar, Grigory M.; Rezounenko, Alexander V.

doi:10.1016/s1631-073x(03)00251-6

Cited by 24 publications

(44 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…is a consequence of Theorem 4, while property H2) follows from the next theorem (see theorems 15 and 16 from [18] and also [16] for more details and proof). m , but in the left half plane, the integer m being in I.…”

Section: We Naturally Put Hmentioning

confidence: 78%

On strong regular stabilizability for linear neutral type systems

Rabah¹

2006

2006 14th Mediterranean Conference on Control and Automation

View full text Add to dashboard Cite

To cite this version:Rabah Rabah, Grigory Sklyar, Aleksandr Rezounenko. On strong regular stabilizability for linear neutral type systems. Journal of Differential Equations, Elsevier, 2008, 245 (3) AbstractThe problem of strong stabilizability of linear systems of neutral type is investigated. We are interested in the case when the system has an infinite sequence of eigenvalues with vanishing real parts. This is the case when the main part of the neutral equation is not assumed to be stable in the classical sense. We discuss the notion of regular strong stabilizability and present an approach to stabilize the system by regular linear controls. The method covers the case of multivariable control and is essentially based on the idea of infinite-dimensional pole assignment proposed in [G.Sklyar, A.Rezounenko, A theorem on the strong asymptotic stability and determination of stabilizing controls. C. R. Acad. Sci. Paris Ser. I Math. 333(2001), No. 8, 807-812]. Our approach is based on the recent results on the Riesz basis of invariant finite-dimensional subspaces and strong stability for neutral type systems presented in [R. Rabah, G.M. Sklyar, A.V. Rezounenko, Stability analysis of neutral type systems in Hilbert space.

show abstract

Section: We Naturally Put Hmentioning

confidence: 78%

On strong regular stabilizability for linear neutral type systems

Rabah¹

2006

2006 14th Mediterranean Conference on Control and Automation

View full text Add to dashboard Cite

show abstract

“…The proofs are based on the construction of a special Riesz basis of Ainvariant subspaces in the space M 2 according to [12] and on the analysis of the properties of some quasi-exponential functions to be a Riesz basis in L 2 (0, T ) depending of the time T [1]. …”

Section: The Main Resultsmentioning

confidence: 99%

“…[12] There exists N 0 large enough such that for any (5) turns into a moment problem with respect to a special collection of quasipolynomials. Analyzing the mentioned moment problem by means of the methods given in [1] we obtain our main results concerning the null-controllability problem.…”

Section: The Choice Of Basismentioning

confidence: 99%

On Exact Controllability of Linear Time Delay Systems of Neutral Type

Rabah¹,

Sklyar

2007

Applications of Time Delay Systems

Self Cite

View full text Add to dashboard Cite

To cite this version:Rabah Rabah, Grigory Sklyar. On exact controllability of linear time delay systems of neutral type. Summary. The problem of exact null controllability is considered for linear neutral type systems with distributed delay. A characterization of this problem is given. The minimal time of controllability is precised. The results are based on the analysis of the Riesz basis property of eigenspaces in Hilbert space. Recent results on the moment problem and properties of exponential families are used.

show abstract

“…For systems (1.1)-(1.3), the resolvent of the operator A allows an explicit representation (see [27,28]). Such a representation is an effective tool for analyzing the exponential stability property since the latter is equivalent to the uniform boundedness of the resolvent on the complex right half-plane.…”

Section: Introductionmentioning

confidence: 99%

Stability and stabilizability of mixed retarded-neutral type systems

2011

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Generalized Riesz basis property in the analysis of neutral type systems

Cited by 24 publications

References 6 publications

On strong regular stabilizability for linear neutral type systems

On strong regular stabilizability for linear neutral type systems

On Exact Controllability of Linear Time Delay Systems of Neutral Type

Stability and stabilizability of mixed retarded-neutral type systems

Contact Info

Product

Resources

About