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

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.
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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