 Previous Article
 CPAA Home
 This Issue

Next Article
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. 
[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] 
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] 
[1] 
Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295313. doi: 10.3934/dcds.2007.18.295 
[2] 
Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021040 
[3] 
Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with statedependent delay. Discrete and Continuous Dynamical Systems  B, 2017, 22 (8) : 31673197. doi: 10.3934/dcdsb.2017169 
[4] 
Tomás Caraballo, Carlos Ogouyandjou, Fulbert Kuessi Allognissode, Mamadou Abdoul Diop. Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process. Discrete and Continuous Dynamical Systems  B, 2020, 25 (2) : 507528. doi: 10.3934/dcdsb.2019251 
[5] 
Jan Čermák, Jana Hrabalová. Delaydependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 45774588. doi: 10.3934/dcds.2014.34.4577 
[6] 
Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with nonlipschitz coefficients. Discrete and Continuous Dynamical Systems  B, 2019, 24 (7) : 32993318. doi: 10.3934/dcdsb.2018321 
[7] 
Stéphane Junca, Bruno Lombard. Stability of neutral delay differential equations modeling wave propagation in cracked media. Conference Publications, 2015, 2015 (special) : 678685. doi: 10.3934/proc.2015.0678 
[8] 
Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discretetime state observations. Discrete and Continuous Dynamical Systems  B, 2017, 22 (1) : 209226. doi: 10.3934/dcdsb.2017011 
[9] 
Fuke Wu, Shigeng Hu. The LaSalletype theorem for neutral stochastic functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 10651094. doi: 10.3934/dcds.2012.32.1065 
[10] 
Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 23692384. doi: 10.3934/cpaa.2020103 
[11] 
Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete and Continuous Dynamical Systems  B, 2013, 18 (6) : 16831696. doi: 10.3934/dcdsb.2013.18.1683 
[12] 
Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete and Continuous Dynamical Systems  B, 2017, 22 (7) : 29232938. doi: 10.3934/dcdsb.2017157 
[13] 
Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary timevarying delay. Discrete and Continuous Dynamical Systems  S, 2011, 4 (3) : 693722. doi: 10.3934/dcdss.2011.4.693 
[14] 
Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete and Continuous Dynamical Systems  S, 2008, 1 (2) : 219223. doi: 10.3934/dcdss.2008.1.219 
[15] 
Tian Zhang, Chuanhou Gao. Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise. Discrete and Continuous Dynamical Systems  B, 2022, 27 (7) : 37253747. doi: 10.3934/dcdsb.2021204 
[16] 
Sibel Senan, Eylem Yucel, Zeynep Orman, Ruya Samli, Sabri Arik. A Novel Lyapunov functional with application to stability analysis of neutral systems with nonlinear disturbances. Discrete and Continuous Dynamical Systems  S, 2021, 14 (4) : 14151428. doi: 10.3934/dcdss.2020358 
[17] 
John A. D. Appleby, John A. Daniels. Exponential growth in the solution of an affine stochastic differential equation with an average functional and financial market bubbles. Conference Publications, 2011, 2011 (Special) : 91101. doi: 10.3934/proc.2011.2011.91 
[18] 
Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha$Norm for a class of partial functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 115135. doi: 10.3934/dcds.2011.30.115 
[19] 
BaoZhu Guo, LiMing Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689694. doi: 10.3934/mbe.2011.8.689 
[20] 
Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations and Control Theory, 2015, 4 (4) : 493505. doi: 10.3934/eect.2015.4.493 
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]