Discrete Lyapunov theory for evolution families
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Abstract
The aim of the paper "Discrete Lyapunov theory for evolution families" is to obtain some necessary and sufficient conditions for uniform exponential dichotomy, in the discrete case for C_0-semigroups, differential systems and abstract evolution families in Hilbert spaces. We prove firstly the discrete versions of some theorems from the continuous case from P. Preda, M. Megan, Exponential dichotomy of strongly discontinuous semigroups, Bull. Austral. Math. Soc. 30(1984), 435-448, P. Preda, M. Megan, Exponential Dichotomy of Evolutionary Processes in Banach Spaces, Czechoslovak Math. J. 35(1985), 312-323 and we use them to obtain the results of Lyapunov type.
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Mihit, C., & Preda, P.
(2015).
Discrete Lyapunov theory for evolution families.
Acta Mathematica Universitatis Comenianae, 84(1), 149-166.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/102/133
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References
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2. C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, vol 70, Providence, RO: American Mathematical Society, 1999.
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4. W.A. Coppel, Dichotomies in Stability Theory, Lect. Notes Math, vol. 629, Springer-Verlag, New-York, 1978.
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7. R. Datko, Uniform Asymptotic Stability of Evolutionary Processes in Banach Space, SIAM J.Math.Anal. 3(1972) 428-445.
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11. J.L. Massera, J. J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New-York, 1966.
12. M. Megan, B. Sasu, A. L. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst. 9 (2) (2003) 383-397.
13. L. Pandolfi, A Lyapunov theorem for semigroups of operators, Systems Control Lett. 15 (1990), 147-151.
14. A. Pazy, On the Applicability of Lyapunov Theorem in Hilbert Space, SIAM J.Math.Anal. 3(1972), 291-294.
15. M. Pinto, Discrete Dichotomies, Computers Math. Applic. Vol. 28, No. 1-3 (1994), 259-270.
16. A. Pogan, C. Preda, P. Preda, A discrete Lyapunov theorem for the exponential stability of evolution families, New York J. Math. 11 (2005), 457-463.
17. P. Preda, M. Megan, Exponential dichotomy of strongly discontinuous semigroups, Bull. Austral. Math. Soc. 30 (1984), 435-448.
18. P. Preda, M. Megan, Exponential Dichotomy of Evolutionary Processes in Banach Spaces, Czechoslovak Math. J. 35 (1985), 312-323.
19. P. Preda, A. Pogan, C. Preda, Discrete admissibility and exponential dichotomy for evolution families, Dynamics of Continuous, Discrete and Impulsive Systems, Math. Analysis 12 (2005) 621-631.
20. K. M. Przyluski, The Lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space, Appl. Math. Optim. 6 (1980), 97-112.
21. K. M. Przyluski, S. Rolewicz, On stability of linear time-varying infinite-dimensional discrete-time systems, Syst. Control Lett. 4 (1984), 307-315.
2. C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs, vol 70, Providence, RO: American Mathematical Society, 1999.
3. C. V. Coffman, J. J. Schaffer, Dichotomies for linear difference equations, Math. Annalen 172, 1967, 139-166.
4. W.A. Coppel, Dichotomies in Stability Theory, Lect. Notes Math, vol. 629, Springer-Verlag, New-York, 1978.
5. J. L. Daleckij, M.G. Krein, Stability of Differential Equations in Banach Space, Amer. Math. Soc., Providence, R.I.1974.
6. R. Datko, Extending a Theorem of Lyapunov to Hilbert Spaces, J. Math. Anal. Appl. 32, 1970, 610-616.
7. R. Datko, Uniform Asymptotic Stability of Evolutionary Processes in Banach Space, SIAM J.Math.Anal. 3(1972) 428-445.
8. K.J. Engel, R. Nagel,One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, 1999.
9. J. P. La Salle, The Stability and Control of Discrete Processes, Springer-Verlag, Berlin, 1990.
10. J.L.Massera, J.J.Schaffer, Linear Differential Equations and Functional Analysis, I. Annals of Mathematics vol.67, No.3, 1958, 517-573.
11. J.L. Massera, J. J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New-York, 1966.
12. M. Megan, B. Sasu, A. L. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst. 9 (2) (2003) 383-397.
13. L. Pandolfi, A Lyapunov theorem for semigroups of operators, Systems Control Lett. 15 (1990), 147-151.
14. A. Pazy, On the Applicability of Lyapunov Theorem in Hilbert Space, SIAM J.Math.Anal. 3(1972), 291-294.
15. M. Pinto, Discrete Dichotomies, Computers Math. Applic. Vol. 28, No. 1-3 (1994), 259-270.
16. A. Pogan, C. Preda, P. Preda, A discrete Lyapunov theorem for the exponential stability of evolution families, New York J. Math. 11 (2005), 457-463.
17. P. Preda, M. Megan, Exponential dichotomy of strongly discontinuous semigroups, Bull. Austral. Math. Soc. 30 (1984), 435-448.
18. P. Preda, M. Megan, Exponential Dichotomy of Evolutionary Processes in Banach Spaces, Czechoslovak Math. J. 35 (1985), 312-323.
19. P. Preda, A. Pogan, C. Preda, Discrete admissibility and exponential dichotomy for evolution families, Dynamics of Continuous, Discrete and Impulsive Systems, Math. Analysis 12 (2005) 621-631.
20. K. M. Przyluski, The Lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space, Appl. Math. Optim. 6 (1980), 97-112.
21. K. M. Przyluski, S. Rolewicz, On stability of linear time-varying infinite-dimensional discrete-time systems, Syst. Control Lett. 4 (1984), 307-315.