Comptes Rendus
no. 1
Ordinary Differential Equations
Generalized Riesz basis property in the analysis of neutral type systems
[Bases généralisées de Riesz pour les systèmes de type neutre]

Présenté par : Pierre-Louis Lions

Rabah Rabah1 ; Grigory M. Sklyar2 ; Alexander V. Rezounenko3
1 IRCCyN UMR 6597, 1, rue de la Noë, PB 92101, 44321 Nantes cedex 3, France
2 Institute of Mathematics, University of Szczecin, 70-451 Szczecin, Wielkopolska 15, Poland
3 Department of Mechanics and Mathematics, Kharkov University, 4 Svobody sqr., Kharkov, 61077, Ukraine
Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 19-24.

On étudie une équation différentielle fonctionnelle de type neutre. Nous considérons le modèle opérationnel dans l'espace de Hilbert M 2 =C n ×L 2 (-1,0;C n ) et montrons qu'il existe dans cet espace une base de Riesz de sous-espaces de dimensions finies invariants par l'opérateur générateur infinitésimal du système. Nous donnons également un exemple précisant qu'il n'existe pas de base de Riesz de sous-espaces propres.

The functional differential equation of neutral type is studied. We consider the corresponding operator model in Hilbert space M 2 =C n ×L 2 (-1,0;C n ) and prove that there exists a sequence of invariant finite-dimensional subspaces which constitute a Riesz basis in M2. We also give an example emphasizing that the generalized eigenspaces do not form a Riesz basis.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00251-6
Rabah Rabah 1 ; Grigory M. Sklyar 2 ; Alexander V. Rezounenko 3

1 IRCCyN UMR 6597, 1, rue de la Noë, PB 92101, 44321 Nantes cedex 3, France
2 Institute of Mathematics, University of Szczecin, 70-451 Szczecin, Wielkopolska 15, Poland
3 Department of Mechanics and Mathematics, Kharkov University, 4 Svobody sqr., Kharkov, 61077, Ukraine 
@article{CRMATH_2003__337_1_19_0,
     author = {Rabah Rabah and Grigory M. Sklyar and Alexander V. Rezounenko},
     title = {Generalized {Riesz} basis property in the analysis of neutral type systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {19--24},
     publisher = {Elsevier},
     volume = {337},
     number = {1},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00251-6},
     language = {en},
}
TY  - JOUR
AU  - Rabah Rabah
AU  - Grigory M. Sklyar
AU  - Alexander V. Rezounenko
TI  - Generalized Riesz basis property in the analysis of neutral type systems
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 19
EP  - 24
VL  - 337
IS  - 1
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00251-6
LA  - en
ID  - CRMATH_2003__337_1_19_0
ER  - 
%0 Journal Article
%A Rabah Rabah
%A Grigory M. Sklyar
%A Alexander V. Rezounenko
%T Generalized Riesz basis property in the analysis of neutral type systems
%J Comptes Rendus. Mathématique
%D 2003
%P 19-24
%V 337
%N 1
%I Elsevier
%R 10.1016/S1631-073X(03)00251-6
%G en
%F CRMATH_2003__337_1_19_0
Rabah Rabah; Grigory M. Sklyar; Alexander V. Rezounenko. Generalized Riesz basis property in the analysis of neutral type systems. Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 19-24. doi : 10.1016/S1631-073X(03)00251-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00251-6/

[1] N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert space, Dover, New York, NY. Transl. from the Russian. Repr. of the 1961 and 1963 transl

[2] R. Bellman; K.L. Cooke Differential-difference equations, Math. Sci. Engrg., 6, Academic Press, New York, 1963 (XVI)

[3] W.E. Brumley On the asymptotic behavior of solutions of differential-difference equations of neutral type, J. Differential Equations, Volume 7 (1970), pp. 175-188

[4] R.F. Curtain; H. Zwart An Introduction to Infinite-Dimensional Linear Systems Theory, Texts Appl. Math., 21, Springer-Verlag, New York, 1995

[5] N. Dunford; J.T. Schwartz Linear Operators, Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space, Interscience, 1963

[6] I.C. Gohberg; M.G. Krein Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, 18, American Mathematical Society, Providence, RI, 1969 (XV, 378 p)

[7] J. Hale; S.M. Verduyn Lunel Theory of Functional Differential Equations, Springer-Verlag, New York, 1993

[8] T. Kato Perturbation Theory for Linear Operators, Springer-Verlag, 1980

[9] V. Kolmanovskii; A. Myshkis Introduction to the Theory and Applications of Functional Differential Equations, Math. Appl., 463, Kluwer Academic, Dordrecht, 1999

[10] D.A. O'Connor; T.J. Tarn On stabilization by state feedback for neutral differential equations, IEEE Trans. Automatic Control, Volume AC-28 (1983) no. 5, pp. 615-618

[11] L. Pandolfi Stabilization of neutral functional differential equations, J. Optimization Theory and Appl., Volume 20 (1976) no. 2, pp. 191-204

[12] R. Rabah, G.M. Sklyar, On a class of strongly stabilizable systems of neutral type, submitted

[13] Y. Yamamoto; S. Ueshima A new model for neutral delay-differential systems, Internat. J. Control, Volume 43 (1986) no. 2, pp. 465-471

[14] Y. Yamamoto Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems, SIAM J. Control Optim., Volume 27 (1989) no. 1, pp. 217-234

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Eigenvalues and eigenvectors assignment for neutral type systems

Katerina Sklyar; Rabah Rabah; Grigory Sklyar

C. R. Math (2013)