On strong regular stabilizability for linear neutral type systems

Abstract

International audienceThe 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.M. Sklyar, A.V. Rezounenko, A theorem on the strong asymptotic stability and determination of stabilizing controls, C. R. Acad. Sci. Paris Ser. I Math. 333 (8) (2001) 807-812]. Our approach is based on the recent results on the Riesz basis of invariant finitedimensional 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, J. Differential Equations 214 (2) (2005) 391-428]

Asymptotic Growth of Solutions of Neutral Type Systems

“…(1). Using Theorem 8 from [16] to both scalar equations we get that for any sequence τ (k) quadratically close to sequenceλ ,…”

“…Operator A 1 generates a C 0 -semigroup and also is of the form (3) with the same matrix A −1 and disturbed A 2 , A 3 (see [16] for more details). Thus assertion follows directly from Theorem 2.1.…”

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

On Asymptotic Estimation of a Discrete Type $$C_0$$ C 0 -Semigroups on Dense Sets: Application to Neutral Type Systems

“…Such a situation may occur for hyperbolic equations or delay equations of neutral type (e.g. Rabah et al [13,14]). …”

“…First we give the proof of Theorem 1.1 preceded by several technical results. Next section is devoted to the analysis of stability of neutral type equations (3) and regular feedback stabilizability of these equations [14]. In the appendix we give two simple statements about complex matrices, which are used in our work.…”