# American Institute of Mathematical Sciences

April  2020, 19(4): 2403-2418. doi: 10.3934/cpaa.2020105

## Sensitivity to small delays of mean square stability for stochastic neutral evolution equations

 1 College of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China 2 Department of Mathematical Sciences, The University of Liverpool, Liverpool, L69 7ZL, U.K

*Corresponding author

Dedicated to Professor Tomás Caraballo on occasion of his 60th Birthday

Received  November 2018 Revised  July 2019 Published  January 2020

Fund Project: Wei Wang is supported by China Scholarship Council (Grant No. 201708120038) and National Natural Science Foundation of China (Grant No. 11601382). Kai Liu is supported by Tianjin Thousand Talents Plan and Xiulian Wang is supported by Project of Tianjin Municipal Education Commission (Grant No. JW1714).

In this work, we are concerned about the mean square exponential stability property for a class of stochastic neutral functional differential equations with small delay parameters. Both distributed and point delays under the neutral term are considered. Sufficient conditions are given to capture the exponential stability in mean square of the stochastic system under consideration. As an illustration, we present some practical systems to show their exponential stability which is not sensitive to small delays in the mean square sense.

Citation: Wei Wang, Kai Liu, Xiulian Wang. Sensitivity to small delays of mean square stability for stochastic neutral evolution equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2403-2418. doi: 10.3934/cpaa.2020105
##### References:
 [1] J. A. Appleby and X. R. Mao, Stochastic stabilisation of functional differential equations, Syst. Control Letters, 54 (2005), 1069-1081.  doi: 10.1016/j.sysconle.2005.03.003. [2] A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, A. K. Peters, Wellesley, Massachusetts, 2005. [3] J. Bierkens, Pathwise stability of degenerate stochastic evolutions, Integr. Equ. Oper. Theory, 23 (2010), 1-27.  doi: 10.1007/s00020-010-1841-4. [4] R. Datko, J. Lagnese and M. Polis, An example on the effect of time delays in boundary feedback of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007. [5] K. Liu, Sensitivity to small delays of pathwise stability for stochastic retarded evolution equations, J. Theoretical Probab., 31 (2018), 1625-1646.  doi: 10.1007/s10959-017-0750-8. [6] K. Liu, Almost sure exponential stability sensitive to small time delay of stochastic neutral functional differential equations, Applied Math. Letters, 77 (2018), 57-63.  doi: 10.1016/j.aml.2017.09.008. [7] K. Liu, Stochastic Stability of Differential Equations in Abstract Spaces, Cambridge University Press, 2019.  doi: 10.1017/9781108653039. [8] K. Liu, Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives, Discrete Cont. Dyn. Sys.-B, 23 (2018), 3915-3934. [9] H. Tanabe, Equations of Evolution, Pitman, New York, 1979.

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##### References:
 [1] J. A. Appleby and X. R. Mao, Stochastic stabilisation of functional differential equations, Syst. Control Letters, 54 (2005), 1069-1081.  doi: 10.1016/j.sysconle.2005.03.003. [2] A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, A. K. Peters, Wellesley, Massachusetts, 2005. [3] J. Bierkens, Pathwise stability of degenerate stochastic evolutions, Integr. Equ. Oper. Theory, 23 (2010), 1-27.  doi: 10.1007/s00020-010-1841-4. [4] R. Datko, J. Lagnese and M. Polis, An example on the effect of time delays in boundary feedback of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007. [5] K. Liu, Sensitivity to small delays of pathwise stability for stochastic retarded evolution equations, J. Theoretical Probab., 31 (2018), 1625-1646.  doi: 10.1007/s10959-017-0750-8. [6] K. Liu, Almost sure exponential stability sensitive to small time delay of stochastic neutral functional differential equations, Applied Math. Letters, 77 (2018), 57-63.  doi: 10.1016/j.aml.2017.09.008. [7] K. Liu, Stochastic Stability of Differential Equations in Abstract Spaces, Cambridge University Press, 2019.  doi: 10.1017/9781108653039. [8] K. Liu, Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives, Discrete Cont. Dyn. Sys.-B, 23 (2018), 3915-3934. [9] H. Tanabe, Equations of Evolution, Pitman, New York, 1979.
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