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Khasminskii-type theorems for stochastic functional differential equations
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 |
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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