Stability and stabilizability of mixed retarded-neutral type systems

Rabah, Rabah ; Sklyar, Grigory Mikhailovitch ; Barkhayev, Pavel Yurevitch

ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 656-692.

Résumé

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.

Détail

DOI : 10.1051/cocv/2011166

Classification : 34K40, 34K20, 93C23, 93D15
Keywords: retarded-neutral type systems, asymptotic non-exponential stability, stabilizability, infinite dimensional systems

Bibliographie

[1] W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306 (1988) 837-852. | Zbl | MR

[2] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups. Math. Nachr. 279 (2006) 1425-1440. | Zbl | MR

[3] R. Bellman and K.L. Cooke, Differential-difference equations. Academic Press, New York (1963). | Zbl | MR

[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. | Zbl | MR

[6] W.E. Brumley, On the asymptotic behavior of solutions of differential-difference equations of neutral type. J. Diff. Equ. 7 (1970) 175-188. | Zbl | MR

[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. | Zbl | MR

[8] R.F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics 21. Springer-Verlag, New York (1995). | Zbl | MR

[9] X. Dusser and R. Rabah, On exponential stabilizability of linear neutral type systems. Math. Probl. Eng. 7 (2001) 67-86. | Zbl | MR

[10] J.K. Hale and S.M.V. Lunel, Introduction to functional-differential equations, Applied Mathematical Sciences 99. Springer-Verlag, New York (1993). | Zbl | MR

[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. | Zbl | MR

[12] D. Henry, Linear autonomous neutral functional differential equations. J. Diff. Equ. 15 (1974) 106-128. | Zbl | MR

[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. Control 33 (1988) 541-549. | Zbl | MR

[14] T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York (1966). | Zbl | MR

[15] V.B. Kolmanovskii and V.R. Nosov, Stability of functional-differential equations, Mathematics in Science and Engineering 180. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1986). | Zbl | MR

[16] L.A. Liusternik and V.J. Sobolev, Elements of functional analysis, Russian Monographs and Texts on Advanced Mathematics and Physics 5. Hindustan Publishing Corp., Delhi (1961). | Zbl | MR

[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). | Zbl | MR

[18] Yu.I. Lyubich and V.Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 88 (1988) 37-42. | Zbl | MR

[19] S.A. Nefedov and F.A. Sholokhovich, A criterion for stabilizability of dynamic systems with finite-dimensional input. Differentsial' nye Uravneniya 22 (1986) 223-228, 364. | Zbl | MR

[20] D.A. O'Connor and T.J. Tarn, On stabilization by state feedback for neutral differential-difference equations. IEEE Trans. Automat. Control 28 (1983) 615-618. | Zbl | MR

[21] L. Pandolfi, Stabilization of neutral functional differential equations. J. Optim. Theory Appl. 20 (1976) 191-204. | Zbl | MR

[22] J.R. Partington and C. Bonnet, H∞ and BIBO stabilization of delay systems of neutral type. Syst. Control Lett. 52 (2004) 283-288. | Zbl | MR

[23] L.S. Pontryagin, On the zeros of some elementary transcendental functions. Amer. Math. Soc. Transl. 1 (1955) 95-110. | Zbl | MR

[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. | Zbl | MR

[25] R. Rabah and G.M. Sklyar, On a class of strongly stabilizable systems of neutral type. Appl. Math. Lett. 18 (2005) 463-469. | Zbl | MR

[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. | Zbl | MR

[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. Paris 337 (2003) 19-24. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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). | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[34] R. Triggiani, On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52 (1975) 383-403. | Zbl | MR

[35] J. Van Neerven, The asymptotic behaviour of semigroups of linear operators, Operator Theory : Advances and Applications 88. Birkhäuser Verlag, Basel (1996). | Zbl | MR

[36] S.M. Verduyn Lunel and D.V. Yakubovich, A functional model approach to linear neutral functional-differential equations. Integr. Equ. Oper. Theory 27 (1997) 347-378. | Zbl | MR

[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). | Zbl | MR

[38] W.M. Wonham, Linear multivariable control, Applications of Mathematics 10. 3th edition, Springer-Verlag, New York (1985). | Zbl | MR