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Sigmoidal approximations of a delay neural lattice model with Heaviside functions
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 
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.
References:
[1] 
J. A. Appleby and X. R. Mao, Stochastic stabilisation of functional differential equations, Syst. Control Letters, 54 (2005), 10691081. doi: 10.1016/j.sysconle.2005.03.003. 
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A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, A. K. Peters, Wellesley, Massachusetts, 2005. 
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J. Bierkens, Pathwise stability of degenerate stochastic evolutions, Integr. Equ. Oper. Theory, 23 (2010), 127. doi: 10.1007/s0002001018414. 
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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), 152156. doi: 10.1137/0324007. 
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K. Liu, Sensitivity to small delays of pathwise stability for stochastic retarded evolution equations, J. Theoretical Probab., 31 (2018), 16251646. doi: 10.1007/s1095901707508. 
[6] 
K. Liu, Almost sure exponential stability sensitive to small time delay of stochastic neutral functional differential equations, Applied Math. Letters, 77 (2018), 5763. 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 highestorder derivatives, Discrete Cont. Dyn. Sys.B, 23 (2018), 39153934. 
[9] 
show all references
References:
[1] 
J. A. Appleby and X. R. Mao, Stochastic stabilisation of functional differential equations, Syst. Control Letters, 54 (2005), 10691081. 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), 127. doi: 10.1007/s0002001018414. 
[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), 152156. doi: 10.1137/0324007. 
[5] 
K. Liu, Sensitivity to small delays of pathwise stability for stochastic retarded evolution equations, J. Theoretical Probab., 31 (2018), 16251646. doi: 10.1007/s1095901707508. 
[6] 
K. Liu, Almost sure exponential stability sensitive to small time delay of stochastic neutral functional differential equations, Applied Math. Letters, 77 (2018), 5763. 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 highestorder derivatives, Discrete Cont. Dyn. Sys.B, 23 (2018), 39153934. 
[9] 
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