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Title: Exponential stability and exponential instability for linear skew-product flows (English)
Author: Megan, Mihail
Author: Sasu, Adina Luminiţa
Author: Sasu, Bogdan
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 129
Issue: 3
Year: 2004
Pages: 225-243
Summary lang: English
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Category: math
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Summary: We give characterizations for uniform exponential stability and uniform exponential instability of linear skew-product flows in terms of Banach sequence spaces and Banach function spaces, respectively. We present a unified approach for uniform exponential stability and uniform exponential instability of linear skew-product flows, extending some stability theorems due to Neerven, Datko, Zabczyk and Rolewicz. (English)
Keyword: linear skew-product flow
Keyword: uniform exponential stability
Keyword: uniform exponential instability
MSC: 34D05
MSC: 34D09
MSC: 34E05
MSC: 34G20
MSC: 37C75
MSC: 47D06
idZBL: Zbl 1080.34538
idMR: MR2092710
DOI: 10.21136/MB.2004.134146
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Date available: 2009-09-24T22:14:32Z
Last updated: 2020-07-29
Stable URL: http://hdl.handle.net/10338.dmlcz/134146
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Reference: [1] Chicone, C.; Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations.Math. Surveys and Monographs, vol. 70, Amer. Math. Soc., 1999. MR 1707332
Reference: [2] Chow, S. N.; Leiva, H.: Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces.J. Differ. Equations 120 (1995), 429–477. MR 1347351, 10.1006/jdeq.1995.1117
Reference: [3] Chow, S. N.; Leiva, H.: Unbounded perturbation of the exponential dichotomy for evolution equations.J. Differ. Equations 129 (1996), 509–531. MR 1404391, 10.1006/jdeq.1996.0125
Reference: [4] Datko, R.: Uniform asymptotic stability of evolutionary processes in Banach spaces.SIAM J. Math. Anal. 3 (1972), 428–445. MR 0320465, 10.1137/0503042
Reference: [5] Henry, D.: Geometric Theory of Semilinear Parabolic Equations.Springer, New York, 1981. Zbl 0456.35001, MR 0610244
Reference: [6] Latushkin, Y.; Montgomery-Smith, S.; Randolph, T.: Evolutionary semigroups and dichotomy of linear skew-product flows on spaces with Banach fibers.J. Differ. Equations 125 (1996), 73–116. MR 1376061, 10.1006/jdeq.1996.0025
Reference: [7] Latushkin, Y.; Schnaubelt, R.: Evolution semigroups, translation algebras and exponential dichotomy of cocycles.J. Differ. Equations 159 (1999), 321–369. MR 1730724, 10.1006/jdeq.1999.3668
Reference: [8] Megan, M.; Sasu, B.; Sasu, A. L.: On nonuniform exponential dichotomy of evolution operators in Banach spaces.Integral Equations Operator Theory 44 (2002), 71–78. MR 1913424, 10.1007/BF01197861
Reference: [9] Megan, M.; Sasu, A. L.; Sasu, B.; Pogan, A.: Exponential stability and unstability of semigroups of linear operators in Banach spaces.Math. Inequal. Appl. 5 (2002), 557–567. MR 1907541
Reference: [10] Megan, M.; Sasu, A. L.; Sasu, B.: On uniform exponential stability of linear skew- product semiflows in Banach spaces.Bull. Belg. Math. Soc. - Simon Stevin 9 (2002), 143–154. MR 1905653, 10.36045/bbms/1102715145
Reference: [11] Megan, M.; Sasu, A. L.; Sasu, B.: Stabilizability and controllability of systems associated to linear skew-product semiflows.Rev. Mat. Complut. 15 (2002), 599–618. MR 1951828, 10.5209/rev_REMA.2002.v15.n2.16932
Reference: [12] Megan, M.; Sasu, A. L.; Sasu, B.: Discrete admissibility and exponential dichotomy for evolution families.Discrete Contin. Dyn. Syst. 9 (2003), 383–397. MR 1952381
Reference: [13] Megan, M.; Sasu, A. L.; Sasu, B.: On uniform exponential dichotomy for linear skew-product semiflows.Bull. Belg. Math. Soc. - Simon Stevin 10 (2003), 1–21. MR 2032321, 10.36045/bbms/1047309409
Reference: [14] Megan, M.; Sasu, B.; Sasu, A. L.: Exponential expansiveness and complete admissibility for evolution families.Accepted in Czechoslovak Math. J. MR 2086730
Reference: [15] Megan, M.; Sasu, A. L.; Sasu, B.: Perron conditions for pointwise and global exponential dichotomy of linear skew-product semiflows.Accepted in Integral Equations Operator Theory.
Reference: [16] Megan, M.; Sasu, A. L.; Sasu, B.: Theorems of Perron type for uniform exponential stability of linear skew-product semiflows.Accepted in Dynam. Contin. Discrete Impuls. Systems.
Reference: [17] Meyer-Nieberg, P.: Banach Lattices.Springer, Berlin, 1991. Zbl 0743.46015, MR 1128093
Reference: [18] Van Neerven, J.: The Asymptotic Behaviour of Semigroups of Linear Operators.Birkhäuser, 1996. Zbl 0905.47001, MR 1409370
Reference: [19] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations.Springer, Berlin, 1983. Zbl 0516.47023, MR 0710486
Reference: [20] Pliss, V. A.; Sell, G. R.: Robustness of exponential dichotomies in infinite-dimensional dynamical systems.J. Dynam. Differ. Equ. 3 (1999), 471–513. MR 1693858, 10.1023/A:1021913903923
Reference: [21] Pliss, V. A.; Sell, G. R.: Perturbations of normally hyperbolic manifolds with applications to the Navier-Stokes equation.J. Differ. Equations 169 (2001), 396–492. MR 1808472, 10.1006/jdeq.2000.3905
Reference: [22] Rolewicz, S.: On uniform $N$-equistability.J. Math. Anal. Appl. 115 (1986), 434–441. Zbl 0597.34064, MR 0836237, 10.1016/0022-247X(86)90006-5
Reference: [23] Sacker, R. J.; Sell, G. R.: Lifting properties in skew-product flows with applications to differential equations.Mem. Am. Math. Soc. 190, Providence, Rhode Island, 1977. MR 0448325
Reference: [24] Sacker, R. J.; Sell, G. R.: Dichotomies for linear evolutionary equations in Banach spaces.J. Differ. Equations 113 (1994), 17–67. MR 1296160, 10.1006/jdeq.1994.1113
Reference: [25] Zabczyk, J.: Remarks on the control of discrete-time distributed parameter systems.SIAM J. Control 12 (1974), 721–735. Zbl 0254.93027, MR 0410506, 10.1137/0312056
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