Asymptotic Stability of autonomous and Non-Autonomous Discrete Linear Inclusions

D. Cheban ; C. Mammana

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2004), pp. 41-52.

Résumé

The article is devoted to the study of absolute asymptotic stability of discrete linear inclusions (both autonomous and non-autonomous) in Banach space. We establish the relation between absolute asymptotic stability, uniform asymptotic stability and uniform exponential stability. It is proved that for compact (completely continuous) discrete linear inclusions these notions of stability are equivalent. We study this problem in the framework of non-autonomous dynamical systems (cocyles).

Bibliographie

[1] Artzrouni M., “On the Convergence of Infinite Products of Matrices”, Linear Algebra and its Applications, 74 (1986), 11–21 | DOI | MR | Zbl

[2] Barabanov N. E., “Lyapunov indicator of discrete inclusions. I; II; III”, Autom. Remote Control, 49 (1988), 152–157 ; 283–287 ; 558–565 | MR | Zbl | MR | Zbl | MR | Zbl

[3] Beyn W.-J., Elsner L., “Infinite Products and Paracontracting Matrices”, The Electronic Journal of Linear Algebra, 2 (1997), 1–8 | MR | Zbl

[4] Bronstein I. U., Nonautonomous Dynamical Systems, Stiintsa, Chişinău, 1984 (in Russian)

[5] Bru R., Elsner L., Neumann M., “Convergence of Infinite Products of Matrices and Inner-Outer Iteration Schemes”, Electronic Transactions on Numerical Analysis, 2 (1994), 183–194 | MR

[6] Cheban D. N., Fakeeh D. S., Global Attractors of Disperse Dynamical Systems, Sigma, Chişinău, 1994

[7] Cheban D. N., Fakeeh D. S., “Global Attractors of Infinite-Dimensional Dynamical Systems, III”, Bulletin of Academy of Sciences of Republic of Moldova, Mathematics, 1995, no. 2-3(18-19), 3–13 | MR | Zbl

[8] Mathematical Notes, 63:1 (1998), 115–126 | MR | Zbl

[9] Cheban D. N., Global Attractors of Nonautonomous Dynamical Systems, State University of Moldova, Chişinău, 2002 (in Russia)

[10] Cheban D. N., Global Attractors of Nonautonomous Dissipative Dynamical Systems, World Scientific (to appear)

[11] Clement Ph., Heijman J., Angement S., C. Van Dujin C., De Pagter P., One-Parameter Semigroups, CWI Monograph, 5, North-Holland, Amsterdam–New York–Oxford–Tokyo, 1987 | MR | Zbl

[12] Daubechies I., Lagarias J. C., “Sets of Matrices All Infnite Products of Which Converge”, Linear Algebra Appl., 161 (1992), 227–263 | DOI | MR | Zbl

[13] Elsner L., Friedland S., “Norm Conditions for Convergence of Infinite Products”, Linear Algebra Appl., 250 (1997), 133–142 | DOI | MR | Zbl

[14] Elsner L., Koltracht I., Neumann M., “On the Convergence of Asynchronous Paracontractions with Applications to Tomographic Reconstruction from Incomplete Data”, Linear Algebra Appl., 130 (1990), 65–82 | DOI | MR | Zbl

[15] Gurvits L., Geometrical Approach to Some Problems of Control, Observation and Lumping, PhD. Thesis, Dept. of Mathematics, Gorky State Univ., USSR, 1985

[16] Gurvits L. N., Zaharin A. M., “System Theoretic and Graph Theoretic Aspects of Markov Chains Lumping”, 5th International Conference on Mathematical Statistics, Vilnius, 1989 | Zbl

[17] Gurvits L. N., Zaharin A. M., Lumpability of Markov Processes and Systems Theory, MTNS, Amsterdam, 1989

[18] Gurvits L., “Stability of Discrete Linear Inclusion”, Linear Algebra and its Applications, 231 (1995), 47–85 | MR | Zbl

[19] Hale J. K., Asymptotic Behaviour of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988

[20] Husemoller D., Fibre Bundles, Springer-Verlag, Berlin–Heidelberg–New York, 1994 | MR

[21] Kelley J., General Topology, D. Van Norstand Company Inc., Toronto–New York–London, 1957 | MR

[22] Kozyakin V. S., “Algebric insolvability of problem of absolute stability of desynchronized systems”, Autom. Rem. Control, 1990, no. 51(6), 754–759 | MR | Zbl

[23] Molchanov A. P., “Lyapunov function for nonlinear discrete-time control systems”, Avtomat. i Telemekh., 6 (1987), 26–35 | MR | Zbl

[24] Opoitsev V. I., “Stability of nonautonomous systems”, Avtomat. i Telemekh., 1986, no. 10, 56–63 | MR

[25] Robinson C., Dynamical Systems: Stabilty, Symbolic Dynamics and Chaos, Studies in Advanced Mathematics, CRC Press, Boca Raton Florida, 1995 | MR

[26] Rota G. C., Strang W. G., “A Note on the Joint Spectral Radius”, Indag. Math., 22 (1960), 379–381 | MR

[27] Sacker R. J., Sell G. R., “Existence of Dichotomies and Invariant Splittings for Linear Differential Systems, I”, Journal of Differential Equations, 15 (1974), 429–458 | DOI | MR | Zbl

[28] Sacker R. J., Sell G. R., “Dichotomies for Linear Evoltionary Equations in Banach Spaces”, Journal of Differential Equations, 113 (1994), 17–67 | DOI | MR | Zbl

[29] Sell G. R., Lectures on Topological Dynamics and Differential Equations, Van Nostrand Reinhold math. studies, 2, Van Nostrand-Reinbold, London, 1971 | MR

[30] Shcherbakov B. A., Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Sţiinţa, Chişinău, 1972 | MR

[31] Vladimirov A., Elsner L., Beyn W.-J., “Stability and paracontractivity of discrete linear inclusion”, Linear Algebra and its Applications, 312 (2000), 125–134 | DOI | MR | Zbl

[32] De Vries J., Elements of Topological Dynamics, Mathematics and Its Applications, 257, Kluwer Academic Publishers, Dordecht–Boston–London, 1993 | MR | Zbl

[33] Wirth F., “On Stability of Infinite-Dimensional Discrete Inclusions”, Journal of Mathematical Systems, Estimations and Control, 8:4 (1998), 507–510 | MR | Zbl

[34] Wirth F., “The generalized spectral radius and external norms”, Linear Algebra end its Applications, 342 (2002), 17–40 | DOI | MR | Zbl