Mean-square random attractors of stochastic delay differential equations with random delay

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August 2013, 18(6): 1715-1734. doi: 10.3934/dcdsb.2013.18.1715

Mean-square random attractors of stochastic delay differential equations with random delay

Fuke Wu ^1, and Peter E. Kloeden ^2,

1.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
2.	Institut für Mathematik, Goethe Universität, D-60054 Frankfurt am Main

Received September 2011 Revised February 2012 Published March 2013

Abstract
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The existence of a random attactor is established for a mean-square random dynamical system (MS-RDS) generated by a stochastic delay equation (SDDE) with random delay for which the drift term is dominated by a nondelay component satisfying a one-sided dissipative Lipschitz condition. It is shown by Razumikhin-type techniques that the solution of this SDDE is ultimately bounded in the mean-square sense and that solutions for different initial values converge exponentially together as time increases in the mean-square sense. Consequently, similar boundedness and convergence properties hold for the MS-RDS and imply the existence of a mean-square random attractor for the MS-RDS that consists of a single stochastic process.

Keywords: Razumikhin-type theorem, random delay, ultimate boundedness., mean-square random dynamical system (MS-RDS), Mean-square random attractor.

Mathematics Subject Classification: 34K50, 37H10, 60H10, 93E0.

Citation: Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715

References:

[1]	L. Arnold, "Stochastic Differential Equations: Theory and Applications," Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
[2]	L. Arnold, Random dynamical systems, in "Dynamical Systems" (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, (1995), 1-43. doi: 10.1007/BFb0095238.
[3]	T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays, Journal of Mathematical Analysis and Applications, 260 (2001), 421-438. doi: 10.1006/jmaa.2000.7464.
[4]	T. Caraballo, P. Marin-Rubío and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, Journal of Differential Equations, 208 (2005), 9-41. doi: 10.1016/j.jde.2003.09.008.
[5]	T. Caraballo, P. E. Kloeden and J. Real, Discretization of asymptoticality stable stationary solutions of delay differential equations with a random stationary delay, Journal of Dynamics and Differential Equations, 18 (2006), 863-880. doi: 10.1007/s10884-006-9022-5.
[6]	T. Caraballo, P. Marin-Rubío and J. Valero, Attractors for differential equations with unbounded delays, Journal of Differential Equations, 239 (2007), 311-342. doi: 10.1016/j.jde.2007.05.015.
[7]	I. Chueshov, "Monotone Random Systems Theory and Applications," Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.
[8]	J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.
[9]	R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980.
[10]	P. E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stochastic Anal. Appl., 28 (2010), 937-945. doi: 10.1080/07362994.2010.515194.
[11]	P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438. doi: 10.1016/j.jde.2012.05.016.
[12]	P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," Mathematical Surveys and Monographs, 176, Amer. Math. Soc., Providence, RI, 2011.
[13]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.
[14]	T. Lorenz, Nonlocal stochastic differential equations: Existence and uniqueness of solutions, Bol. Soc. Esp. Mat. Apl. SeMA, 51 (2010), 99-107.
[15]	X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994.
[16]	X. Mao, "Stochatic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chirchester, 2008.
[17]	X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23 (2005), 1045-1069. doi: 10.1080/07362990500118637.
[18]	L. Montestruque and P. Antsaklis, Stability of model-based networked control systems with time-varying transmission times, IEEE Transaction on Automatical Control, 49 (2004), 1562-1572. doi: 10.1109/TAC.2004.834107.
[19]	J. Nilsson, B. Bernhardsson and B. Wittenmark, Stochastic analysis and control of real-time systems with random time delays, Automatica J. IFAC, 34 (1998), 57-64. doi: 10.1016/S0005-1098(97)00170-2.
[20]	L. Schenato, Optimal estimation in networked control systems subject to random delay and packet drop, IEEE Transaction on Automatical Control, 53 (2008), 1311-1317. doi: 10.1109/TAC.2008.921012.

show all references

References:

[1]	L. Arnold, "Stochastic Differential Equations: Theory and Applications," Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
[2]	L. Arnold, Random dynamical systems, in "Dynamical Systems" (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, (1995), 1-43. doi: 10.1007/BFb0095238.
[3]	T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays, Journal of Mathematical Analysis and Applications, 260 (2001), 421-438. doi: 10.1006/jmaa.2000.7464.
[4]	T. Caraballo, P. Marin-Rubío and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, Journal of Differential Equations, 208 (2005), 9-41. doi: 10.1016/j.jde.2003.09.008.
[5]	T. Caraballo, P. E. Kloeden and J. Real, Discretization of asymptoticality stable stationary solutions of delay differential equations with a random stationary delay, Journal of Dynamics and Differential Equations, 18 (2006), 863-880. doi: 10.1007/s10884-006-9022-5.
[6]	T. Caraballo, P. Marin-Rubío and J. Valero, Attractors for differential equations with unbounded delays, Journal of Differential Equations, 239 (2007), 311-342. doi: 10.1016/j.jde.2007.05.015.
[7]	I. Chueshov, "Monotone Random Systems Theory and Applications," Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.
[8]	J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.
[9]	R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980.
[10]	P. E. Kloeden and T. Lorenz, Stochastic differential equations with nonlocal sample dependence, Stochastic Anal. Appl., 28 (2010), 937-945. doi: 10.1080/07362994.2010.515194.
[11]	P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438. doi: 10.1016/j.jde.2012.05.016.
[12]	P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," Mathematical Surveys and Monographs, 176, Amer. Math. Soc., Providence, RI, 2011.
[13]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.
[14]	T. Lorenz, Nonlocal stochastic differential equations: Existence and uniqueness of solutions, Bol. Soc. Esp. Mat. Apl. SeMA, 51 (2010), 99-107.
[15]	X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994.
[16]	X. Mao, "Stochatic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chirchester, 2008.
[17]	X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23 (2005), 1045-1069. doi: 10.1080/07362990500118637.
[18]	L. Montestruque and P. Antsaklis, Stability of model-based networked control systems with time-varying transmission times, IEEE Transaction on Automatical Control, 49 (2004), 1562-1572. doi: 10.1109/TAC.2004.834107.
[19]	J. Nilsson, B. Bernhardsson and B. Wittenmark, Stochastic analysis and control of real-time systems with random time delays, Automatica J. IFAC, 34 (1998), 57-64. doi: 10.1016/S0005-1098(97)00170-2.
[20]	L. Schenato, Optimal estimation in networked control systems subject to random delay and packet drop, IEEE Transaction on Automatical Control, 53 (2008), 1311-1317. doi: 10.1109/TAC.2008.921012.

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