Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay. (English) Zbl 1223.93096
Summary: This paper establishes the existence-and-uniqueness theorem of neutral stochastic functional differential equations with infinite delay and examines the almost sure stability of this solution with general decay rate. This result may be used to examine almost sure robust stability. To illustrate our idea more carefully, we discuss a scalar stochastic integro-differential equation of neutral type and its asymptotic stability, including the exponential stability and the polynomial stability.
MSC:
| 93D09 | Robust stability |
| 93E03 | Stochastic systems in control theory (general) |
| 34K40 | Neutral functional-differential equations |
| 93C10 | Nonlinear systems in control theory |
| 34K50 | Stochastic functional-differential equations |
Keywords:
general decay stability; neutral stochastic functional differential equations; infinite delay; global solutionsReferences:
| [1] | Kolmanovskii, V. B.; Nosov, V. R., Stability of Functional Differential Equations (1986), Academic Press: Academic Press New York · Zbl 0593.34070 |
| [2] | Haussmann, U. G., Asymptotic stability of the linear Itô equation in finite dimensions, J. Math. Anal. Appl., 65, 219-235 (1978) · Zbl 0385.93051 |
| [3] | Ichikawa, A., Stability of semilinear stochastic evolutions, J. Math. Anal. Appl., 90, 12-44 (1982) · Zbl 0497.93055 |
| [4] | Mao, X., Stability of Stochastic Differential Equations with Respect to Semimartingale (1991), Wiley: Wiley New York · Zbl 0724.60059 |
| [5] | Mao, X., Robustness of stability of nonlinear systems with stochastic delay perturbations, System Control Lett., 19, 391-400 (1992) · Zbl 0763.93064 |
| [6] | Appleby, J. A.D.; Riedle, M., Almost sure asymptotic stability of stochastic Volterra integro-differential equations with fading perburbations, Stoch. Anal. Appl., 24, 813-826 (2006) · Zbl 1121.60070 |
| [7] | Mao, X., Stability of stochastic integro-differential equations, Stoch. Anal. Appl., 18, 1005-1017 (2000) · Zbl 0969.60068 |
| [8] | Mao, X.; Riedle, M., Mean square stability of stochastic Volterra integro-differential equations, System Control Lett., 55, 459-465 (2006) · Zbl 1129.34332 |
| [9] | Hale, J. K.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer: Springer Berlin · Zbl 0787.34002 |
| [10] | Mao, X., Stochastic Differential Equations and Applications (1997), Horwood: Horwood Chichester · Zbl 0874.60050 |
| [11] | Arnold, L., Stochastic Differential Equations: Theory and Applications (1972), Wiley: Wiley New York |
| [12] | Hu, Y.; Wu, F.; Huang, C., Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay, Automatica, 2577-2584 (2009) · Zbl 1368.93765 |
| [13] | Mao, X., Exponential stability in mean square of neutral stochastic differential functional equations, System Control Lett., 26, 245-251 (1995) · Zbl 0877.93133 |
| [14] | Mao, X., Razumikhin-type theorems on exponential stability of neutral stochastic functional differential, SIAM J. Math. Anal., 28, 389-401 (1997) · Zbl 0876.60047 |
| [15] | Mao, X., A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268, 125-142 (2002) · Zbl 0996.60064 |
| [16] | Randjelović, J.; Janković, S., On the \(p\) th moment exponential stability criteria of neutral stochastic functional differential equations, J. Math. Anal. Appl., 326, 266-280 (2007) · Zbl 1115.60065 |
| [17] | Deng, F.; Luo, Q.; Mao, X.; Pang, S., Noise suppresses or expresses exponential growth, System Control Lett., 57, 262-270 (2008) · Zbl 1157.93515 |
| [18] | Mao, X., Almost sure polynomial stability for a class of stochastic differential equations, Quart. J. Math. Oxford, 43, 339-348 (1992) · Zbl 0765.60058 |
| [19] | Caraballo, T.; Garrido-Atienza, M. J.; Real, J., Stochastic stabilization of differential systems with general decay rate, System Control Lett., 48, 397-406 (2003) · Zbl 1157.93537 |
| [20] | Liu, K.; Mao, X., Large time decay behavior of dynamical equations with random perturbation features, Stoch. Anal. Appl., 19, 295-327 (2001) · Zbl 0997.93095 |
| [21] | Kallenberg, O., Foundations of Modern Probability (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0892.60001 |
| [22] | Liptser, R. Sh.; Shiryaev, A. N., Theory of Martingales (1989), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0482.60030 |
| [23] | Fang, S.; Zhang, T., A study of a class of stochastic differential equations with non-Lipschitzian coefficients, Probab. Theory Related Fields, 132, 356-390 (2005) · Zbl 1081.60043 |
| [24] | Mao, X., Exponential Stability of Stochastic Differential Equations (1994), Dekker: Dekker New York · Zbl 0851.93074 |
| [25] | Ren, Y.; Xia, N., Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput., 210, 72-79 (2009) · Zbl 1167.34389 |
| [26] | Wei, F.; Wang, K., The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 331, 516-531 (2007) · Zbl 1121.60064 |
| [27] | R.Z. Khasminskii, Stochastic stability of differential equations, Sijthoff and Noordhoff, Alphen a/d Rijn, 1981.; R.Z. Khasminskii, Stochastic stability of differential equations, Sijthoff and Noordhoff, Alphen a/d Rijn, 1981. |
| [28] | Mao, X.; Rassias, M. J., Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23, 1045-1069 (2005) · Zbl 1082.60055 |
| [29] | Higham, D. J.; Mao, X.; Yuan, C., Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 45, 592-607 (2007) · Zbl 1144.65005 |
| [30] | Wu, F.; Xu, Y., Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70, 641-657 (2009) · Zbl 1197.34164 |
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.
