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September  2014, 13(5): 2095-2113. doi: 10.3934/cpaa.2014.13.2095

## Stability of delay evolution equations with stochastic perturbations

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 2 Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev str., 163-a, Donetsk, 83015

Received  December 2012 Revised  February 2013 Published  June 2014

The investigation of stability for hereditary systems is often related to the construction of Lyapunov functionals. The general method of Lyapunov functionals construction, which was proposed by V.Kolmanovskii and L.Shaikhet, is used here to investigate the stability of stochastic delay evolution equations, in particular, for stochastic partial differential equations. This method had already been successfully used for functional-differential equations, for difference equations with discrete time, and for difference equations with continuous time. It is shown that the stability conditions obtained for stochastic 2D Navier-Stokes model with delays are essentially better than the known ones.
Citation: Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095
##### References:
 [1] T. Caraballo, M. J. Garrido-Atienza and J. Real, Asymptotic stability of nonlinear stochastic evolution equations, Stoch. Anal. Appl., 21 (2003), 301-327. doi: 10.1081/SAP-120019288. [2] T. Caraballo and K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch. Anal. Appl., 15 (1999), 743-763. doi: 10.1080/07362999908809633. [3] T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay properties, Proc. Roy. Soc. Lond. A, 456 (2000), 1775-1802. doi: 10.1098/rspa.2000.0586. [4] T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl., 334 (2007), 1130-1145. doi: 10.1016/j.jmaa.2007.01.038. [5] H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays, Proc. Indian Acad. Sci (Math. Sci), 122 (2012), 283-295. doi: 10.1007/s12044-012-0071-x. [6] V. Kolmanovskii and L. Shaikhet, A method of Lyapunov functionals construction for stochastic differential equations of neutral type, Differentialniye uravneniya, 31 (2002), 691-716 (in Russian). Translation in: Differential Equations, 31 (1996), 1819-1825. [7] V. Kolmanovskii and L. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results, Mathematical and Computer Modelling, 36 (1995), 1851-1857. [8] J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. anal. Appl., 342 (2008), 753-760. doi: 10.1016/j.jmaa.2007.11.019. [9] E. Pardoux, Equations aux dérivées partielles stochastiques nonlinéaires monotones, Ph.D thesis, Université Paris Sud, 1975. [10] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Encyclopedia of mathematics and its applications, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. [11] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, Dordrecht, Heidelberg, New York, 2011. doi: 10.1007/978-0-85729-685-6. [12] L. Shaikhet, Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems, Theory of Stochastic Processes, 2 (1996), 248-259. [13] L. Wan and J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statistics and Probability Letters, 78 (2008), 490-498. doi: 10.1016/j.spl.2007.08.003. [14] M. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier-Stokes equations with time delay, Appl. Math. J. Chinese Univ., 24 (2009), 493-500 (in Chinese).

show all references

##### References:
 [1] T. Caraballo, M. J. Garrido-Atienza and J. Real, Asymptotic stability of nonlinear stochastic evolution equations, Stoch. Anal. Appl., 21 (2003), 301-327. doi: 10.1081/SAP-120019288. [2] T. Caraballo and K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch. Anal. Appl., 15 (1999), 743-763. doi: 10.1080/07362999908809633. [3] T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay properties, Proc. Roy. Soc. Lond. A, 456 (2000), 1775-1802. doi: 10.1098/rspa.2000.0586. [4] T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl., 334 (2007), 1130-1145. doi: 10.1016/j.jmaa.2007.01.038. [5] H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays, Proc. Indian Acad. Sci (Math. Sci), 122 (2012), 283-295. doi: 10.1007/s12044-012-0071-x. [6] V. Kolmanovskii and L. Shaikhet, A method of Lyapunov functionals construction for stochastic differential equations of neutral type, Differentialniye uravneniya, 31 (2002), 691-716 (in Russian). Translation in: Differential Equations, 31 (1996), 1819-1825. [7] V. Kolmanovskii and L. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results, Mathematical and Computer Modelling, 36 (1995), 1851-1857. [8] J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. anal. Appl., 342 (2008), 753-760. doi: 10.1016/j.jmaa.2007.11.019. [9] E. Pardoux, Equations aux dérivées partielles stochastiques nonlinéaires monotones, Ph.D thesis, Université Paris Sud, 1975. [10] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Encyclopedia of mathematics and its applications, Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223. [11] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer, London, Dordrecht, Heidelberg, New York, 2011. doi: 10.1007/978-0-85729-685-6. [12] L. Shaikhet, Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems, Theory of Stochastic Processes, 2 (1996), 248-259. [13] L. Wan and J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statistics and Probability Letters, 78 (2008), 490-498. doi: 10.1016/j.spl.2007.08.003. [14] M. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier-Stokes equations with time delay, Appl. Math. J. Chinese Univ., 24 (2009), 493-500 (in Chinese).
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