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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 & 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,, \emph{Stoch. Anal. Appl.}, 21 (2003), 301.  doi: 10.1081/SAP-120019288.  Google Scholar

[2]

T. Caraballo and K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{Stoch. Anal. Appl.}, 15 (1999), 743.  doi: 10.1080/07362999908809633.  Google Scholar

[3]

T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay properties,, \emph{Proc. Roy. Soc. Lond. A}, 456 (2000), 1775.  doi: 10.1098/rspa.2000.0586.  Google Scholar

[4]

T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations,, \emph{J. Math. Anal. Appl.}, 334 (2007), 1130.  doi: 10.1016/j.jmaa.2007.01.038.  Google Scholar

[5]

H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays,, \emph{Proc. Indian Acad. Sci (Math. Sci)}, 122 (2012), 283.  doi: 10.1007/s12044-012-0071-x.  Google Scholar

[6]

V. Kolmanovskii and L. Shaikhet, A method of Lyapunov functionals construction for stochastic differential equations of neutral type,, \emph{Differentialniye uravneniya}, 31 (2002), 691.   Google Scholar

[7]

V. Kolmanovskii and L. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results,, \emph{Mathematical and Computer Modelling}, 36 (1995), 1851.   Google Scholar

[8]

J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{J. Math. anal. Appl.}, 342 (2008), 753.  doi: 10.1016/j.jmaa.2007.11.019.  Google Scholar

[9]

E. Pardoux, Equations aux dérivées partielles stochastiques nonlinéaires monotones,, Ph.D thesis, (1975).   Google Scholar

[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.  Google Scholar

[11]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations,, Springer, (2011).  doi: 10.1007/978-0-85729-685-6.  Google Scholar

[12]

L. Shaikhet, Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems,, \emph{Theory of Stochastic Processes}, 2 (1996), 248.   Google Scholar

[13]

L. Wan and J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory,, \emph{Statistics and Probability Letters}, 78 (2008), 490.  doi: 10.1016/j.spl.2007.08.003.  Google Scholar

[14]

M. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier-Stokes equations with time delay,, \emph{Appl. Math. J. Chinese Univ.}, 24 (2009), 493.   Google Scholar

show all references

References:
[1]

T. Caraballo, M. J. Garrido-Atienza and J. Real, Asymptotic stability of nonlinear stochastic evolution equations,, \emph{Stoch. Anal. Appl.}, 21 (2003), 301.  doi: 10.1081/SAP-120019288.  Google Scholar

[2]

T. Caraballo and K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{Stoch. Anal. Appl.}, 15 (1999), 743.  doi: 10.1080/07362999908809633.  Google Scholar

[3]

T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay properties,, \emph{Proc. Roy. Soc. Lond. A}, 456 (2000), 1775.  doi: 10.1098/rspa.2000.0586.  Google Scholar

[4]

T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations,, \emph{J. Math. Anal. Appl.}, 334 (2007), 1130.  doi: 10.1016/j.jmaa.2007.01.038.  Google Scholar

[5]

H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays,, \emph{Proc. Indian Acad. Sci (Math. Sci)}, 122 (2012), 283.  doi: 10.1007/s12044-012-0071-x.  Google Scholar

[6]

V. Kolmanovskii and L. Shaikhet, A method of Lyapunov functionals construction for stochastic differential equations of neutral type,, \emph{Differentialniye uravneniya}, 31 (2002), 691.   Google Scholar

[7]

V. Kolmanovskii and L. Shaikhet, Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results,, \emph{Mathematical and Computer Modelling}, 36 (1995), 1851.   Google Scholar

[8]

J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{J. Math. anal. Appl.}, 342 (2008), 753.  doi: 10.1016/j.jmaa.2007.11.019.  Google Scholar

[9]

E. Pardoux, Equations aux dérivées partielles stochastiques nonlinéaires monotones,, Ph.D thesis, (1975).   Google Scholar

[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.  Google Scholar

[11]

L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations,, Springer, (2011).  doi: 10.1007/978-0-85729-685-6.  Google Scholar

[12]

L. Shaikhet, Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems,, \emph{Theory of Stochastic Processes}, 2 (1996), 248.   Google Scholar

[13]

L. Wan and J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory,, \emph{Statistics and Probability Letters}, 78 (2008), 490.  doi: 10.1016/j.spl.2007.08.003.  Google Scholar

[14]

M. Wei and T. Zhang, Exponential stability for stochastic 2D-Navier-Stokes equations with time delay,, \emph{Appl. Math. J. Chinese Univ.}, 24 (2009), 493.   Google Scholar

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