Stability and stabilizability of mixed retarded-neutral type systems∗

Rabah Rabah; Grigory Mikhailovitch Sklyar; Pavel Yurevitch Barkhayev

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 656-692
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Rabah, Rabah, Sklyar, Grigory Mikhailovitch, and Barkhayev, Pavel Yurevitch. "Stability and stabilizability of mixed retarded-neutral type systems∗." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 656-692. <http://eudml.org/doc/277824>.

@article{Rabah2012,
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. },
author = {Rabah, Rabah, Sklyar, Grigory Mikhailovitch, Barkhayev, Pavel Yurevitch},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Retarded-neutral type systems; asymptotic non-exponential stability; stabilizability; infinite dimensional systems; retarded-neutral type systems},
language = {eng},
month = {11},
number = {3},
pages = {656-692},
publisher = {EDP Sciences},
title = {Stability and stabilizability of mixed retarded-neutral type systems∗},
url = {http://eudml.org/doc/277824},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Rabah, Rabah
AU - Sklyar, Grigory Mikhailovitch
AU - Barkhayev, Pavel Yurevitch
TI - Stability and stabilizability of mixed retarded-neutral type systems∗
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 656
EP - 692
AB - 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.
LA - eng
KW - Retarded-neutral type systems; asymptotic non-exponential stability; stabilizability; infinite dimensional systems; retarded-neutral type systems
UR - http://eudml.org/doc/277824
ER -

References

top
  1. W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc.306 (1988) 837–852.  
  2. A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups. Math. Nachr.279 (2006) 1425–1440.  
  3. R. Bellman and K.L. Cooke, Differential-difference equations. Academic Press, New York (1963).  
  4. C. Bonnet, A.R. Fioravanti and J.R. Partington, On the stability of neutral linear systems with multiple commensurated delays, in IFAC Workshop on Control of Distributed Parameter Systems. Toulouse (2009) 195–196. IFAC/LAAS-CNRS.  
  5. A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Ann.347 (2010) 455–478.  
  6. W.E. Brumley, On the asymptotic behavior of solutions of differential-difference equations of neutral type. J. Diff. Equ.7 (1970) 175–188.  
  7. J.A. Burns, T.L. Herdman and H.W. Stech, Linear functional-differential equations as semigroups on product spaces. SIAM J. Math. Anal.14 (1983) 98–116.  
  8. R.F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics21. Springer-Verlag, New York (1995).  
  9. X. Dusser and R. Rabah, On exponential stabilizability of linear neutral type systems. Math. Probl. Eng.7 (2001) 67–86.  
  10. J.K. Hale and S.M.V. Lunel, Introduction to functional-differential equations, Applied Mathematical Sciences99. Springer-Verlag, New York (1993).  
  11. J.K. Hale and S.M.V. Lunel, Strong stabilization of neutral functional differential equations. IMA J. Math. Control Inf.19 (2002) 5–23. Special issue on analysis and design of delay and propagation systems.  
  12. D. Henry, Linear autonomous neutral functional differential equations. J. Diff. Equ.15 (1974) 106–128.  
  13. C.A. Jacobson and C.N. Nett, Linear state-space systems in infinite-dimensional space : the role and characterization of joint stabilizability/detectability. IEEE Trans. Automat. Control33 (1988) 541–549.  
  14. T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band132. Springer-Verlag New York, Inc., New York (1966).  
  15. V.B. Kolmanovskii and V.R. Nosov, Stability of functional-differential equations, Mathematics in Science and Engineering180. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1986).  
  16. L.A. Liusternik and V.J. Sobolev, Elements of functional analysis, Russian Monographs and Texts on Advanced Mathematics and Physics5. Hindustan Publishing Corp., Delhi (1961).  
  17. Z.-H. Luo, B.-Z. Guo and O. Morgul, Stability and stabilization of infinite dimensional systems with applications. Communications and Control Engineering Series, Springer-Verlag London Ltd., London (1999).  
  18. Yu.I. Lyubich and V.Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces. Studia Math.88 (1988) 37–42.  
  19. S.A. Nefedov and F.A. Sholokhovich, A criterion for stabilizability of dynamic systems with finite-dimensional input. Differentsial’ nye Uravneniya22 (1986) 223–228, 364.  
  20. D.A. O’Connor and T.J. Tarn, On stabilization by state feedback for neutral differential-difference equations. IEEE Trans. Automat. Control28 (1983) 615–618.  
  21. L. Pandolfi, Stabilization of neutral functional differential equations. J. Optim. Theory Appl.20 (1976) 191–204.  
  22. J.R. Partington and C. Bonnet, H∞ and BIBO stabilization of delay systems of neutral type. Syst. Control Lett.52 (2004) 283–288.  
  23. L.S. Pontryagin, On the zeros of some elementary transcendental functions. Amer. Math. Soc. Transl.1 (1955) 95–110.  
  24. R. Rabah and G.M. Sklyar, Strong stabilizability for a class of linear time delay systems of neutral type. Mat. Fiz. Anal. Geom.11 (2004) 314–330.  
  25. R. Rabah and G.M. Sklyar, On a class of strongly stabilizable systems of neutral type. Appl. Math. Lett.18 (2005) 463–469.  
  26. R. Rabah and G.M. Sklyar, The analysis of exact controllability of neutral-type systems by the moment problem approach. SIAM J. Control Optim.46 (2007) 2148–2181.  
  27. R. Rabah, G.M. Sklyar and A.V. Rezounenko, Generalized Riesz basis property in the analysis of neutral type systems. C. R. Math. Acad. Sci. Paris337 (2003) 19–24.  
  28. R. Rabah, G.M. Sklyar and A.V. Rezounenko, Stability analysis of neutral type systems in Hilbert space. J. Diff. Equ.214 (2005) 391–428.  
  29. R. Rabah, G.M. Sklyar and A.V. Rezounenko, On strong regular stabilizability for linear neutral type systems. J. Diff. Equ.245 (2008) 569–593.  
  30. G.M. Sklyar, Lack of maximal asymptotics for some linear equations in a Banach space. Dokl. Math.81 (2010) 265–267. Extended version to appear in Taiwanese Journal of Mathematics (2011).  
  31. G.M. Sklyar and A.V. Rezounenko, Stability of a strongly stabilizing control for systems with a skew-adjoint operator in Hilbert space. J. Math. Anal. Appl.254 (2001) 1–11.  
  32. G.M. Sklyar and A.V. Rezounenko, A theorem on the strong asymptotic stability and determination of stabilizing controls. C. R. Acad. Sci. Paris Sér. I Math.333 (2001) 807–812.  
  33. G.M. Sklyar and V.Ya. Shirman, On asymptotic stability of linear differential equation in Banach space. Teor. Funktsiĭ Funktsional. Anal. i Prilozhen.37 (1982) 127–132.  
  34. R. Triggiani, On the stabilizability problem in Banach space. J. Math. Anal. Appl.52 (1975) 383–403.  
  35. J. van Neerven, The asymptotic behaviour of semigroups of linear operators, Operator Theory : Advances and Applications88. Birkhäuser Verlag, Basel (1996).  
  36. S.M. Verduyn Lunel and D.V. Yakubovich, A functional model approach to linear neutral functional-differential equations. Integr. Equ. Oper. Theory27 (1997) 347–378.  
  37. V.V. Vlasov, Spectral problems that arise in the theory of differential equations with delay. Sovrem. Mat. Fundam. Napravl.1 (2003) 69–83 (electronic).  
  38. W.M. Wonham, Linear multivariable control, Applications of Mathematics10. 3th edition, Springer-Verlag, New York (1985).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.