Global attracting set and stability of stochastic neutral partial functional differential equations with impulses. (English) Zbl 1250.93124
Summary: A class of stochastic neutral partial functional differential equations with impulses is investigated. To this end, we first establish a new impulsive-integral inequality, which improve the inequality established by Chen H. B. Chen [”Impulsive-integral inequality and exponential stability for stochastic partial differential equation with delays”. Statist. Probab. Lett. 80, 50–56 (2010; Zbl 1177.93075)]. By using the new inequality, we obtain the global attracting set of stochastic neutral partial functional differential equations with impulses. Especially, sufficient conditions ensuring the exponential \(p\)-stability of the mild solution of the considered equations are obtained. Our results can generalize and improve the existing works. An example is given to demonstrate the main results.
MSC:
| 93E15 | Stochastic stability in control theory |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
Citations:
Zbl 1177.93075References:
| [1] | Anguraj, A.; Vinodkumar, A., Existence, uniqueness and stability of impulsive stochastic partial neutral functional differential equations with infinite delays, J. Appl. Math. Inform., 28, 3-4, 739-751 (2010) · Zbl 1295.93071 |
| [2] | Anguraj, A.; Vinodkumar, A., Existence, uniqueness and stability results of random impulsive semilinear differential systems, Nonlinear Anal. Hybrid Syst., 4, 475-483 (2010) · Zbl 1208.34093 |
| [3] | Bernfeld, S. R.; Corduneanu, C.; Ignatyev, A. O., On the stability of invariant sets of functional differential equations, Nonlinear Anal., 55, 641-656 (2003) · Zbl 1044.34027 |
| [4] | Caraballo, T., Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics, 33, 27-47 (1990) · Zbl 0723.60074 |
| [5] | Caraballo, T.; Liu, K., Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch. Anal. Appl., 17, 743-763 (1999) · Zbl 0943.60050 |
| [6] | Caraballo, T.; Real, J.; Taniguchi, T., The exponential stability of neutral stochastic delay partial differential equations, Discrete Contin. Dyn. Syst., 18, 2-3, 259-313 (2007) · Zbl 1125.60059 |
| [7] | Chen, H. B., Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statist. Probab. Lett., 80, 50-56 (2010) · Zbl 1177.93075 |
| [8] | Cui, J.; Yan, L. T.; Sun, X. C., Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps, Statist. Probab. Lett., 81, 1970-1977 (2011) · Zbl 1227.60085 |
| [9] | Govindan, T. E., Exponential stability in mean-square of parabolic quasilinear stochastic delay evolution equations, Stoch. Anal. Appl., 17, 443-461 (1999) · Zbl 0940.60076 |
| [10] | Govindan, T. E., Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics, 77, 139-154 (2005) · Zbl 1115.60064 |
| [11] | Liao, X. X.; Luo, Q.; Zeng, Z. G., Positive invariant and global exponential attractive sets of neural networks with time-varying delays, Neurocomputing, 71, 513-518 (2008) |
| [12] | Liu, K., Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stochastic, 63, 1-26 (1998) · Zbl 0947.93037 |
| [13] | Liu, K., Stability of Infinite Dimensional Stochastic Differential Equations With Applications (2006), Chapman & Hall/CRC: Chapman & Hall/CRC New York · Zbl 1085.60003 |
| [14] | Liu, K., The fundamental solution and its role in the optimal control of infinite dimensional neutral systems, Appl. Math. Optim., 60, 1-38 (2009) · Zbl 1211.49030 |
| [15] | Liu, K.; Xia, X., On the exponential stability in mean square of neutral stochastic functional differential equations, Systems Control Lett., 37, 207-215 (1999) · Zbl 0948.93060 |
| [16] | Luo, J. W., Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342, 753-760 (2008) · Zbl 1157.60065 |
| [17] | Luo, J. W., Stability of stochastic partial differential equations with infinite delays, J. Comput. Appl. Math., 222, 364-371 (2008) · Zbl 1151.60336 |
| [18] | Luo, J. W., Exponential stability for stochastic neutral partial functional differential equations, J. Math. Anal. Appl., 355, 414-425 (2009) · Zbl 1165.60024 |
| [19] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag New York. · Zbl 0516.47023 |
| [20] | Sakthivel, R.; Luo, J., Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl., 356, 1-6 (2009) · Zbl 1166.60037 |
| [21] | Sakthivel, R.; Luo, J., Asymptotic stability of nonlinear impulsive stochastic differential equations, Statist. Probab. Lett., 78, 1219-1223 (2009) · Zbl 1166.60316 |
| [22] | Sakthivel, R.; Ren, Y.; Kim, H., Asymptotic stability of second-order neutral stochastic differential equations, J. Math. Phys., 51, 052701 (2010) · Zbl 1310.35248 |
| [23] | Taniguchi, T.; Liu, K.; Truman, A., Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic function differential equation in Hilbert spaces, J. Differential Equations, 181, 72-91 (2002) · Zbl 1009.34074 |
| [24] | Wan, L.; Duan, J., Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statist. Probab. Lett., 78, 490-498 (2008) · Zbl 1141.37030 |
| [25] | Xu, D. Y., Invariant and attracting sets of Volterra difference equations with delays, Comput. Math. Appl., 45, 1311-1317 (2003) · Zbl 1065.39021 |
| [26] | Xu, D. Y.; Zhao, H. Y., Invariant set and attractivity of nonlinear differential equations with delays, Appl. Math. Lett., 15, 321-325 (2002) · Zbl 1023.34066 |
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.
