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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

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 & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715
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L. Arnold, "Stochastic Differential Equations: Theory and Applications," Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

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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.  Google Scholar

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J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," Mathematical Surveys and Monographs, 176, Amer. Math. Soc., Providence, RI, 2011.  Google Scholar

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Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.  Google Scholar

[14]

T. Lorenz, Nonlocal stochastic differential equations: Existence and uniqueness of solutions, Bol. Soc. Esp. Mat. Apl. SeMA, 51 (2010), 99-107.  Google Scholar

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X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994.  Google Scholar

[16]

X. Mao, "Stochatic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chirchester, 2008.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

show all references

References:
[1]

L. Arnold, "Stochastic Differential Equations: Theory and Applications," Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

I. Chueshov, "Monotone Random Systems Theory and Applications," Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[8]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," Mathematical Surveys and Monographs, 176, Amer. Math. Soc., Providence, RI, 2011.  Google Scholar

[13]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.  Google Scholar

[14]

T. Lorenz, Nonlocal stochastic differential equations: Existence and uniqueness of solutions, Bol. Soc. Esp. Mat. Apl. SeMA, 51 (2010), 99-107.  Google Scholar

[15]

X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994.  Google Scholar

[16]

X. Mao, "Stochatic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chirchester, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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