Lyapunov type equation for discrete exponential trichotomies
-
1919
Downloads
-
3437
Views
Authors
Davor Dragičević
- School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia.
Abstract
For a nonautonomous dynamics obtained by a sequence of linear operators acting on an arbitrary Hilbert space, we give a
complete characterization of the notion of a uniform exponential trichotomy in terms of what can be considered to be a discrete
version of the Lyapunov equation. We then use this characterization to study the stability of exponential trichotomies under
small linear and nonlinear perturbations.
Share and Cite
ISRP Style
Davor Dragičević, Lyapunov type equation for discrete exponential trichotomies, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 4, 2001--2017
AMA Style
Dragičević Davor, Lyapunov type equation for discrete exponential trichotomies. J. Nonlinear Sci. Appl. (2017); 10(4):2001--2017
Chicago/Turabian Style
Dragičević, Davor. "Lyapunov type equation for discrete exponential trichotomies." Journal of Nonlinear Sciences and Applications, 10, no. 4 (2017): 2001--2017
Keywords
- Exponential trichotomies
- robustness
- perturbations.
MSC
References
-
[1]
A. I. Alonso, J.-L. Hong, R. Obaya, Exponential dichotomy and trichotomy for difference equations, Comput. Math. Appl., 38 (1999), 41–49.
-
[2]
L. Barreira, D. Dragičević, C. Valls, Exponential dichotomies with respect to a sequence of norms and admissibility, Internat. J. Math., 23 (2014), 20 pages.
-
[3]
L. Barreira, D. Dragičević, C. Valls, Admissibility and nonuniform exponential trichotomies, Regul. Chaotic Dyn., 20 (2015), 49–62.
-
[4]
L. Barreira, D. Dragičević, C. Valls, From one-sided dichotomies to two-sided dichotomies, Discrete Contin. Dyn. Syst., 35 (2015), 2817–2844.
-
[5]
L. Barreira, D. Dragičević, C. Valls, Characterization of nonuniform exponential trichotomies for flows, J. Math. Anal. Appl., 434 (2016), 376–400.
-
[6]
L. Barreira, D. Dragičević, C. Valls, Nonuniform exponential dichotomies and Lyapunov functions, Regul. Chaotic Dyn., 22 (2017), 197–209
-
[7]
L. Barreira, C. Valls, Center manifolds for nonuniformly partially hyperbolic diffeomorphisms, J. Math. Pures Appl., 84 (2005), 1693–1715.
-
[8]
L. Barreira, C. Valls, Smooth center manifolds for nonuniformly partially hyperbolic trajectories, J. Differential Equations, 237 (2007), 307–342.
-
[9]
L. Barreira, C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst., 22 (2008), 509–528.
-
[10]
L. Barreira, C. Valls, Stability of nonautonomous differential equations, Lecture Notes in Mathematics, Springer, Berlin (2008)
-
[11]
L. Barreira, C. Valls, Lyapunov sequences for exponential trichotomies, Nonlinear Anal., 72 (2010), 192–203.
-
[12]
L. Barreira, C. Valls , Robustness of dichotomies and trichotomies for difference equations, Port. Math., 71 (2014), 277–300.
-
[13]
, , , (),
-
[14]
M. I. Brin, J. B. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 177–212.
-
[15]
J. Carr , Applications of centre manifold theory, Applied Mathematical Sciences, Springer-Verlag, New York-Berlin (1981)
-
[16]
C. Chicone, Y. Latushkin, Center manifolds for infinite-dimensional nonautonomous differential equations, J. Differential Equations, 141 (1997), 356–399.
-
[17]
S.-N. Chow, W.-S. Liu, Y.-F. Yi, Center manifolds for invariant sets, Special issue in celebration of Jack K. Hale’s 70th birthday, Part 2, Atlanta, GA/Lisbon, (1998), J. Differential Equations, 168 (2000), 355–385.
-
[18]
S.-N. Chow, W.-S. Liu, Y.-F. Yi, Center manifolds for smooth invariant manifolds, Trans. Amer. Math. Soc., 352 (2000), 5179–5211.
-
[19]
S.-N. Chow, Y.-F. Yi, Center manifold and stability for skew-product flows, J. Dynam. Differential Equations, 6 (1994), 543–582.
-
[20]
J. L. Daleckiĭ, M. G. Kreĭn, Stability of solutions of differential equations in Banach space, Translated from the Russian by S. Smith, Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I. (1974)
-
[21]
R. Datko, Extending a theorem of A. M. Liapunov to Hilbert space, J. Math. Anal. Appl., 32 (1970), 610–616.
-
[22]
D. Dragičević, A version of a theorem of R. Datko for stability in average, Systems Control Lett., 96 (2016), 1–6.
-
[23]
D. Dragičević, C. Preda, Lyapunov theorems for exponential dichotomies in Hilbert spaces, Internat. J. Math., 27 (2016), 13 pages.
-
[24]
A. Ducrot, P. Magal, O. Seydi, Persistence of exponential trichotomy for linear operators: a Lyapunov-Perron approach, J. Dynam. Differential Equations, 28 (2016), 93–126.
-
[25]
S. Elaydi, O. Hájek, Exponential trichotomy of differential systems, J. Math. Anal. Appl., 129 (1988), 362–374.
-
[26]
S. Elaydi, K. Janglajew, Dichotomy and trichotomy of difference equations, J. Differ. Equations Appl., 3 (1998), 417–448.
-
[27]
K. J. Engel, R. Nagel , One-parameter semigroups for linear evolution equations, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, Graduate Texts in Mathematics, Springer-Verlag, New York (2000)
-
[28]
M. Faieghi, S. K. Mousavi Mashhadi, D. Baleanu, Sampled-data nonlinear observer design for chaos synchronization: a Lyapunov-based approach, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2444–2453.
-
[29]
T. Gallay, A center-stable manifold theorem for differential equations in Banach spaces, Comm. Math. Phys., 152 (1993), 249–268.
-
[30]
J. A. Goldstein, On the operator equation AX + XB = Q, Proc. Amer. Math. Soc., 70 (1978), 31–34.
-
[31]
J. K. Hale, L. T. Magalhães, W. M. Oliva, Dynamics in infinite dimensions, With an appendix by Krzysztof P. Rybakowski, Second edition, Applied Mathematical Sciences, Springer-Verlag, New York (2002)
-
[32]
B. Hasselblatt, Y. Pesin, Partially hyperbolic dynamical systems, Handbook of dynamical systems, Elsevier B. V., Amsterdam, 1B (2005), 1–55.
-
[33]
D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin- New York (1981)
-
[34]
Y.-X. Jiang, F.-F. Liao, Lyapunov functions for general nonuniform trichotomy with different growth rates, Electron. J. Differential Equations, 2014 (2014), 17 pages.
-
[35]
J. López-Fenner, M. Pinto, (h, k)-trichotomies and asymptotics of nonautonomous difference systems, Comput. Math. Appl., 33 (1997), 105–124.
-
[36]
G. Papaschinopoulos, On exponential trichotomy of linear difference equations, Appl. Anal., 40 (1991), 89-109.
-
[37]
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York, (1983),
-
[38]
V. Q. Phóng, The operator equation AX - XB = C with unbounded operators A and B and related abstract Cauchy problems, Math. Z., 208 (1991), 567–588.
-
[39]
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.
-
[40]
C. Preda, P. Preda, F. Bătăran, An extension of a theorem of R. Datko to the case of (non)uniform exponential stability of linear skew-product semiflows, J. Math. Anal. Appl., 425 (2015), 1148–1154.
-
[41]
C. Preda, P. Preda, A. Craciunescu, A version of a theorem of R. Datko for nonuniform exponential contractions, J. Math. Anal. Appl., 385 (2012), 572–581.
-
[42]
A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906–920.
-
[43]
A. L. Sasu, B. Sasu, Input-output admissibility and exponential trichotomy of difference equations, J. Math. Anal. Appl., 380 (2011), 17–32.
-
[44]
B. Sasu, A. L. Sasu, Nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces, Nonlinear Anal., 74 (2011), 5097–5110.
-
[45]
B. Sasu, A. L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 33 (2013), 3057–3084.
-
[46]
A. L. Sasu, B. Sasu, A Zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems, Appl. Math. Comput., 245 (2014), 447–461.
-
[47]
A. L. Sasu, B. Sasu, Discrete admissibility and exponential trichotomy of dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 2929–2962.
-
[48]
A. L. Sasu, B. Sasu, Admissibility and exponential trichotomy of dynamical systems described by skew-product flows, J. Differential Equations, 260 (2016), 1656–1689.
-
[49]
G. R. Sell, Y.-C. You, Dynamics of evolutionary equations, Applied Mathematical Sciences, Springer-Verlag, New York (2002)
-
[50]
A. Vanderbauwhede, G. Iooss, Center manifold theory in infinite dimensions, Dynamics reported: expositions in dynamical systems, Dynam. Report. Expositions Dynam. Systems (N.S.), Springer, Berlin, 1 (1992), 125–163.
-
[51]
A. Vanderbauwhede, S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal., 72 (1987), 209–224.
-
[52]
G.-C. Wu, D. Baleanu, Jacobian matrix algorithm for Lyapunov exponents of the discrete fractional maps, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 95–100.
