Title:
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Exponential stability and exponential instability for linear skew-product flows (English) |
Author:
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Megan, Mihail |
Author:
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Sasu, Adina Luminiţa |
Author:
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Sasu, Bogdan |
Language:
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English |
Journal:
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Mathematica Bohemica |
ISSN:
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0862-7959 (print) |
ISSN:
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2464-7136 (online) |
Volume:
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129 |
Issue:
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3 |
Year:
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2004 |
Pages:
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225-243 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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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:
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linear skew-product flow |
Keyword:
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uniform exponential stability |
Keyword:
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uniform exponential instability |
MSC:
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34D05 |
MSC:
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34D09 |
MSC:
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34E05 |
MSC:
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34G20 |
MSC:
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37C75 |
MSC:
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47D06 |
idZBL:
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Zbl 1080.34538 |
idMR:
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MR2092710 |
DOI:
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10.21136/MB.2004.134146 |
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Date available:
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2009-09-24T22:14:32Z |
Last updated:
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2020-07-29 |
Stable URL:
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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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