×

Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay. (English) Zbl 1368.93765

Summary: We regard the stochastic functional differential equation with infinite delay \[ dx(t)= f(x_t)dt+ g(x_t)\,dw(t) \] as the result of the effects of stochastic perturbation to the deterministic functional differential equation \[ \dot x (t)= f(x_t), \] where \(x_t= x_t(\theta)\in C((-\infty,0];\mathbb{R}^n)\) is defined by \(x_t(\theta )=x(t+\theta )\), \(\theta\in (-\infty ,0]\). We assume that the deterministic system with infinite delay is exponentially stable.
In this paper, we characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable.

MSC:

93E15 Stochastic stability in control theory
93C73 Perturbations in control/observation systems
93D09 Robust stability
34K20 Stability theory of functional-differential equations
34K50 Stochastic functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Appleby, J. A.D.; Mao, X., Stochastic stabilisation of functional differential equations, System & Control Letters, 54, 1069-1081 (2005) · Zbl 1129.34330
[2] Appleby, J. A.D.; Mao, X.; Rodkina, A., Stabilization and destabilization of nonlinear differential equations by noise, IEEE Transactions on Automatic Control, 53, 683-691 (2008) · Zbl 1367.93692
[3] Arnold, L., Stochastic differential equations: Theory and applications (1972), Wiley: Wiley New York
[4] Boulanger, C., Stabilization of a class of nonlinear stochastic systems, Nonlinear Analysis, 41, 277-286 (2000) · Zbl 0962.93088
[5] Caraballo, T.; Garrido-Atienza, M.; Real, J., Stochastic stabilization of differential systems with general decay rate, System & Control Letters, 48, 397-406 (2003) · Zbl 1157.93537
[6] Fang, S.; Zhang, T., A study of a class of stochastic differential equations with non-Lipschitzian coefficients, Probability Theory and Related Fields, 132, 356-390 (2005) · Zbl 1081.60043
[7] Hu, L.; Mao, X., Almost sure exponential stabilisation of stochastic systems by state-feedback control, Automatica, 44, 465-471 (2008) · Zbl 1283.93303
[8] Iserles, A., On the generalized pantograph functional-differential equation, European Journal of Applied Mathematics, 4, 1-38 (1993) · Zbl 0767.34054
[9] Iserles, A., Exact and discretized stability of the pantograph equation, Applied Numerical Mathematics, 24, 295-308 (1997) · Zbl 0880.65058
[10] Kallenberg, O., Foundations of modern probability (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0892.60001
[11] Khasminskii, R. Z., Stochastic stability of differential equations (1981), Sijthoff and Noordhoff, Alphen a/d Rijn · Zbl 1259.60058
[12] Liptser, R. Sh.; Shiryaev, A. N., Theory of martingale (1989), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0654.60035
[13] Lu, C.; Tsai, J.; Jong, G.; Su, T., An LMI-based approach for robust stabilization of uncertainty stochastic systems with time-varying delays, IEEE Transactions on Automatic Control, 48, 286-289 (2003) · Zbl 1364.93713
[14] Mao, X., Robustness of stability of nonliear systems with stochastic delay perturbations, Systems & Control Letters, 19, 391-400 (1992) · Zbl 0763.93064
[15] Mao, X., Exponential stability of stochastic differential equations (1994), Dekker · Zbl 0851.93074
[16] Mao, X., Robustness of exponential stability of stochastic differential equations, IEEE Transactions on Automatic Control, 41, 442-447 (1996) · Zbl 0851.93074
[17] Mao, X., Stochatic differential equations and applications (1997), Horwood Publishing: Horwood Publishing Chichester, England, UK
[18] Mao, X.; Koroleva, N.; Rodkina, A., Robust stability of uncertain stochastic differential delay equations, Systme & Control Letters, 35, 325-336 (1998) · Zbl 0909.93054
[19] Mao, X.; Rassias, M. J., Khasminskii-type theorems for stochastic differential delay equations, Stochastic Analysis and Applications, 23, 1045-1069 (2005) · Zbl 1082.60055
[20] Mao, X., Stability and stabilisation of stochastic delay differential equations, IET Control Theory & Applications, 6, 1551-1566 (2007)
[21] Mao, X.; Yin, G.; Yuan, C., Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43, 264-273 (2007) · Zbl 1111.93082
[22] Scheutzow, M., Stabilization and destabilization by noise in the plane, Stochastic Analysis and Applications, 11, 97-113 (1993) · Zbl 0766.60072
[23] Shen, Y.; Luo, Q.; Mao, X., The improved LaSalle-type theorems for stochastic functional differential equations, Journal of Mathematical Analysis and Applications, 318, 134-154 (2006) · Zbl 1090.60059
[24] Teng, Z., Permanence and stability in non-autonomous logistic systems with infinite delay, Dynamical Systems: An International Journal, 17, 187-202 (2002) · Zbl 1035.34086
[25] Wei, F.; Wang, K., The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 331, 516-531 (2007) · Zbl 1121.60064
[26] Yuan, C.; Mao, X., Robust stability and controllability of stochastic differential delay equations with Markovian switching, Automatica, 40, 343-354 (2004) · Zbl 1040.93069
[27] Zhang, J.; Suda, Y.; Iwasa, T., Absolutely exponential stability of a class of neural networks with unbounded delay, Neural Networks, 17, 391-397 (2004) · Zbl 1074.68057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.