Abstract
For linear stationary systems, the infinite dimensional framework allows one to distinguish different notions of stability: weak, strong or exponential. The purpose of this chapler is to investigate the problem of strong stability, i.e. asymptotic non-exponential stability for linear systems of neutral type in order to use this characterization in the study of the stabilizability problem for this type of systems. An important tool in this investigation is the Riesz basis property of generalized eigenspaces of the neutral system, because that the generalized eigenvectors do not form, in general, a Riesz basis. This allows one to describe more precisely asymptotic non-exponential stability of neutral systems. For a particular case, conditions of strong stabilizability of neutral type systems are given with a feedback law without derivative of the delayed state.
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References
Akhiezer N. I., Glazman I. M. (1993) Theory of linear operators in Hilber1 space. Translated from the Russian and with a preface by Merlynd Nestell. Reprint of the 1961 and 1963 translations. Two volumes. Dover Publications New York. Theory of linear operators in Hilbcn space. Vol. I, II. Transl. from the 3rd Russian ed. Monographs and Studies in Mathematics, 9, 10. Edinburgh. Boston-London-Melbourne Pitman Advanced Publishing Program. XXXII, 552 p.
Arendt W., Bally C. J. (1988) Tauberian theorems and stability of one-parameter semi-grouos. Trans. Amer. Math. Soc. 306: 837–852
Batty C. J., Phong V. Q. (1990) Stability of individual elements under one-parameter semigroups. Trans. Amer. Math. Soc. 322: 805–818.
Brumley W. E. (1970) On the asymptotic behavior of solutions of differential-difference equations of neutral type. J. Differential Equations 7, 175–188.
Cunain R. F., Zwart H. (1995) An introduction to infinite-dimensional linear systems theory. Springer-Verlag New York.
Dickmann O, van Gils S, Verduyn Lunel S. M., Walther H-O (1995) Delay cquations. Functional, complex, and nonlinear analysis. Applied Mathematical Sciences, 110. Springer-Verlag New York.
Gohberg I. C., Krein M. G. (1969) Introduction to the theory of linear nonselfadjoint operators (English) Translations of Mathematical Monographs. 18. Providence, RI AMS, XV, 378 p.
Hale J., Verduyn Lunel S. M. (1993) Theory of functional differential equations. Springer-Verlag New York.
Kato T (1980) Perturbation theory for linear operators. Springer Verlag.
Kolmanovskii V, Myshkis A (1999) Introduction to the theory and applications of functional differential equations., Mathematics and its Applications (Dordrecht). 463. Dordrecht Kluwer Academic Publishers.
Korobov V. I., Sklyar G. M. (1984) Strong stabilizability of contrnctive systems in Hilbert space. Differentsial’nye Uravn. 20: 1862–1869.
Krabs W., Sklyar G. M. (2002) On Controllability of Linear Vibrations. Nova Science Publ. Huntigton, N. Y.
Levan N., Rigby I. (1979) Strong stabilizability of linear contractive systems in Banach space. SIAM J. Control, 17: 23–35.
Lyubich Yu. I., Phong V. Q. (1988) Asymptotic stability of linear differential equation in Banach space. Studia Math. 88:37–42.
O’Connor D. A, Tam T. J. (1983) On stabilization by state feedback for neutral differential equations. IEEE Transactions on Automatic Control. Vol. AC-28n. 5, 615–618.
Oostvenn J (1999) Strongly stabilizable infinite dimensional systems. Ph.D. Thesis. University of Groningen.
Pandolfi L. (1976) Stabilization of neutral functional differential equations. J. Optimization Theory and Appl. 20n. 2, 191–204.
Rabah R., Sklyar G. M., On a class of strongly stabilizable systems of neutral type, (submitted).
Rabah R., Sklyar G. M., and Rezounenko A. V (2003) Generalized Riesz basis property in the analysis of neutral type systems, Comptes Rendus de l’ Académie des Sciences, Série Mathématiques. To appear. (See also the extended version, Preprint RI02-10, IRCCyN, Names, France).
Sklyar G.M., Shirmao V.Ya. (1982) On Asymptotic Stability of Linear Differential Equation in Banach Space. Teor, Funk., Funkt. Analiz. Prilozh. 37: 127–132
Sklyar G, Rezounenko A. (2001) A theorem on the strong asymptotic stability and determination of stabilizing control. C.R.Acad. Sci. Paris, Ser. I. 333:807–812.
Slemrod M (1973) A note on complcte controllability and stabilizability for linear control systems in Hilbert space. SIAM J. Control 12:500–508.
Sz.-Nagy, B., Foias, C. (1970) Harmonic Analysis of Operators on Hilbert Space. Budapest: Akadémiai Kiadó; Amsterdam-London North-Holland Publishing Company. XIII, 387 p.
van Neerven J. (1996) The asymptotic behaviour of scmigroups of linear operators, in “Operator Theory: Advances and Applications”, Vol. 88. BaselBirkhause
Yamamoto Y, Ueshima S (1986) A new model for neutral delay-differential systems. Internal. J. Control 43(2):46S–471.
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Rabah, R., Sklyar, G.M., Rezounenko, A.V. (2004). On Strong Stability and Stabilizability of Linear Systems of Neutral Type. In: Niculescu, SI., Gu, K. (eds) Advances in Time-Delay Systems. Lecture Notes in Computational Science and Engineering, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18482-6_19
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