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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Brumley, W.E.: On the asymptotic behavior of solutions of differential-difference equations of neutral type. J. Differential Equations 7, 175–188 (1970)
Burns, J.A., Herdman, T.L., Stech, H.W.: Linear functional-differential equations as semigroups on product spaces. SIAM J. Math. Anal. 14(1), 98–116 (1983)
Curtain, R.F., Zwart, H.: An introduction to infinite-dimensional linear systems theory. Springer, New York (1995)
Dusser, X., Rabah, R.: On exponential stabilizability of linear neutral systems. Math. Probl. Eng. 7(1), 67–86 (2001)
Loiseau, J.J., Cardelli, M., Dusser, X.: Neutral-type time-delay systems that are not formally stable are not BIBO stabilizable. Special issue on analysis and design of delay and propagation systems. IMA J. Math. Control Inform. 19(1-2), 217–227 (2002)
Hale, J.K., Verduyn Lunel, S.M.: Theory of functional differential equations. Springer, New York (1993)
Hale, J.K., Verduyn Lunel, S.M.: Strong stabilization of neutral functional differential equations. IMA Journal of Mathematical Control and Information 19(1-2), 5–23 (2002)
Kato, T.: Perturbation theory for linear operators. Springer, Heidelberg (1980)
Korobov, V.I., Sklyar, G.M.: Strong stabilizability of contractive systems in Hilbert space. Differentsial’nye Uravn. 20, 1862–1869 (1984)
Verduyn Lunel, S.M., Yakubovich, D.V.: A functional model approach to linear neutral functional differential equations. Integral Equa. Oper. Theory 27, 347–378 (1997)
Nefedov, S.A., Sholokhovich, F.A.: A criterion for stabilizability of dynamic systems with finite-dimensional input. Differentsial’nye Uravneniya 22(2), 223–228 (1986) (Russian); English translation in the same journal edited by Plenum, New York pp. 163–166
O’Connor, D.A., Tarn, T.J.: On stabilization by state feedback for neutral differential equations. IEEE Transactions on Automatic Control 28(5), 615–618 (1983)
Pandolfi, L.: Stabilization of neutral functional differential equations. J. Optimization Theory and Appl. 20(2), 191–204 (1976)
Rabah, R., Karrakchou, J.: On exact controllablity and complete stabilizability for linear systems in Hilbert spaces. Applied Mathematics Letters 10(1), 35–40 (1997)
Rabah, R., Sklyar, G.M.: The analysis of exact controllability of neutral type systems by the moment problem approach. SIAM J. Control and Optimization 46(6), 2148–2181 (2007)
Rabah, R., Sklyar, G.M., Rezounenko, A.V.: Generalized Riesz basis property in the analysis of neutral type systems. C. R. Math. Acad. Sci. Paris 337(1), 19–24 (2003)
Rabah, R., Sklyar, G.M., Rezounenko, A.V.: On strong stability and stabilizability of systems of neutral type. In: Advances in time-delay systems. LNCSE, vol. 38, pp. 257–268. Springer, Heidelberg (2004)
Rabah, R., Sklyar, G.M., Rezounenko, A.V.: Stability analysis of neutral type systems in Hilbert space. J. Differential Equations 214(2), 391–428 (2005)
Rabah, R., Sklyar, G.M., Rezounenko, A.V.: On strong regular stabilizability for linear neutral type systems. J. Differential Equations 245(3), 569–593 (2008)
Sklyar, G.M.: The problem of the perturbation of an element of a Banach algebra by a right ideal and its application to the question of the stabilization of linear systems in Banach spaces, vol. 230, pp. 32–35. Vestn. Khar’kov. Univ. (1982) (Russian)
Sklyar, G., Rezounenko, A.: A theorem on the strong asymptotic stability and determination of stabilizing control. C.R. Acad. Sci. Paris, Ser. I. 333, 807–812 (2001)
Sklyar, G.M., Rezounenko, A.V.: Strong asymptotic stability and constructing of stabilizing controls. Mat. Fiz. Anal. Geom. 10(4), 569–582 (2003)
Sklyar, G.M., Ya, S.V.: On Asymptotic Stability of Linear Differential Equation in Banach Space. Teor, Funk. Funkt. Analiz. Prilozh. 37, 127–132 (1982)
Slemrod, M.: A note on complete controllability and stabilizability for linear control systems in Hilbert space. SIAM J. Control 12, 500–508 (1973)
van Neerven, J.: The asymptotic behaviour of semigroups of linear operators. In: Operator Theory: Advances and Applications, p. 88. Birkhäuser, Basel (1996)
Wonham, W.M.: Linear multivariable control. A geometric approach, 3rd edn. Springer, New York (1985)
Zabczyk, J.: Mathematical Control Theory: an introduction. Birkäuser, Boston (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Rabah, R., Sklyar, G.M., Rezounenko, A.V. (2009). On Pole Assignment and Stabilizability of Neutral Type Systems. In: Loiseau, J.J., Michiels, W., Niculescu, SI., Sipahi, R. (eds) Topics in Time Delay Systems. Lecture Notes in Control and Information Sciences, vol 388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02897-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-02897-7_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02896-0
Online ISBN: 978-3-642-02897-7
eBook Packages: EngineeringEngineering (R0)