×

Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. (English) Zbl 1456.35043

Summary: A pullback random attractor is called forward controllable if its time-component is semi-continuous to a compact set in the future, and the minimum among all such compact limit-sets is called a forward controller. The existence of a forward controller closely relates to the forward compactness of the attractor, which is further argued by the forward-pullback asymptotic compactness of the system. The abstract results are applied to the non-autonomous stochastic sine-Gordon equation on an unbounded domain. The existence of a forward compact attractor is proved, which leads to the existence of a forward controller. The measurability of the attractor is proved by considering two different universes.

MSC:

35B41 Attractors
35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35R60 PDEs with randomness, stochastic partial differential equations
37L55 Infinite-dimensional random dynamical systems; stochastic equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93B05 Controllability
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] P. Bates; K. Lu; B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289, 32-50 (2014) · Zbl 1364.34113
[2] B. Birnir; R. Grauer, An explicit description of the global attractor of the damped and driven sine-Gordon equation, Comm. Math. Phys., 162, 539-590 (1994) · Zbl 0805.35122
[3] Z. Brzezniak; B. Goldys; Q. T. Le Gia, Random Attractors for the Stochastic Navier-Stokes Equations on the 2D Unit Sphere, J. Math. Fluid Mech., 20, 227-253 (2018) · Zbl 1390.35260
[4] T. Caraballo; M. J. Garrido-Atienza; J. Lopez-de-la-Cruz, Dynamics of some stochastic chemostat models with multiplicative noise, Commun. Pure Appl. Anal., 16, 1893-1914 (2017) · Zbl 1367.34056
[5] V. V. Chepyzhov; M. I. Vishik; W. L. Wendland, On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 15, 27-38 (2005) · Zbl 1067.35017
[6] I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, 1779. Springer-Verlag, Berlin, 2002. · Zbl 1023.37030
[7] I. Chueshov; P. E. Kloeden; M. Yang, Synchronization in coupled stochastic sine-Gordon wave model, Discrete Contin. Dyn. Syst. Ser. B, 21, 2969-2990 (2016) · Zbl 1352.37187
[8] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of Amer Math Soc, 195 (2008). · Zbl 1151.37059
[9] I. Chueshov; I. Lasiecka; D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Differential Equations, 21, 269-314 (2009) · Zbl 1173.35025
[10] I. Chueshov; A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrodinger-Boussinesq equations, Evol. Equ. Control Theory, 1, 57-80 (2012) · Zbl 1282.35320
[11] H. Crauel; P. E. Kloeden; M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11, 301-314 (2011) · Zbl 1235.60076
[12] H. Cui; P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265, 6166-6186 (2018) · Zbl 1408.37036
[13] H. Cui; P. E. Kloeden; F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374, 21-34 (2018) · Zbl 1392.37042
[14] H. Cui; J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263, 1225-1268 (2017) · Zbl 1377.37038
[15] H. Cui; J. A. Langa; Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30, 1873-1898 (2018) · Zbl 1401.37088
[16] X. M. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl, 24, 767-793 (2006) · Zbl 1103.37053
[17] X. M. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216, 63-76 (2004) · Zbl 1065.37057
[18] A. H. Gu and P. E. Kloeden, Asymptotic behavior of a nonautonomous p-Laplacian lattice system, Internat. J. Bifur. Chaos, 26 (2016), 9 pp. · Zbl 1352.34014
[19] X. M. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376, 481-493 (2011) · Zbl 1209.60038
[20] D. S. Jorge; A. Marcio; V. Narciso, Long-time Dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6, 437-470 (2017) · Zbl 1366.35185
[21] P. E. Kloeden; T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144, 259-268 (2016) · Zbl 1362.37045
[22] P. E. Kloeden; J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425, 911-918 (2015) · Zbl 1310.35203
[23] P. E. Kloeden; J. Simsen; M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445, 513-531 (2017) · Zbl 1349.35443
[24] P. E. Kloeden; M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Equ. Appl., 22, 513-525 (2016)
[25] I. Lasiecka; R. Triggiani, Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers, Nonlinear Anal., 121, 424-446 (2015) · Zbl 1320.35261
[26] D. S. Li; B. X. Wang; X. H. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262, 1575-1602 (2017) · Zbl 1360.60125
[27] F. Z. Li; Y. R. Li; R. H. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38, 3663-3685 (2018) · Zbl 1396.35036
[28] Y. R. Li; A. H. Gu; J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258, 504-534 (2015) · Zbl 1306.37091
[29] Y. R. Li; L. B. She; R. H. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459, 1106-1123 (2018) · Zbl 1382.37082
[30] Y. R. Li; L. B. She; J. Y. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23, 1535-1557 (2018) · Zbl 1403.37087
[31] Y. R. Li; R. H. Wang; L. B. She, Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations, Evol. Equ. Control Theory, 7, 617-637 (2018) · Zbl 1405.35007
[32] Y. R. Li; R. H. Wang; J. B. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. Ser. B, 22, 2569-2586 (2017) · Zbl 1367.37057
[33] Y. R. Li; S. Yang, Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise, Commun. Pure Appl. Anal., 18, 1155-1175 (2019) · Zbl 1411.35048
[34] Y. R. Li; J. B. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21, 1203-1223 (2016) · Zbl 1348.37114
[35] Z. W. Shen; S. F. Zhou; W. X. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248, 1432-1457 (2010) · Zbl 1190.60047
[36] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. · Zbl 0871.35001
[37] B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst, 34, 269-300 (2014) · Zbl 1277.35068
[38] B. X. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on \(R^3\), Trans. Amer. Math. Soc, 363, 3639-3663 (2011) · Zbl 1230.37095
[39] B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253, 1544-1583 (2012) · Zbl 1252.35081
[40] S. L. Wang; Y. R. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Physica D, 382, 46-57 (2018) · Zbl 1415.37073
[41] X. H. Wang; K. N. Lu; B. X. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28, 1309-1335 (2016) · Zbl 1353.34101
[42] Z. J. Wang; Y. N. Liu, Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped sine-Gordon equation on unbounded domains, Comput. Math. Appl, 73, 1445-1460 (2017) · Zbl 1370.34110
[43] J. Y. Yin; Y. R. Li; A. H. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74, 744-758 (2017) · Zbl 1383.35032
[44] J. Y. Yin; Y. R. Li; H. Y. Cui, Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450, 1180-1207 (2017) · Zbl 1375.35053
[45] S. F. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263, 2247-2279 (2017) · Zbl 1364.37158
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.