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Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion
1. | School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, China |
2. | Department of Mathematics, Anhui Normal University, Wuhu 241000, China |
3. | School of Mathematics, Southeast University, Nanjing 211189, China |
This paper investigates the issue of weighted exponentially input to state stability (EISS, in short) of stochastic coupled systems on networks with time-varying delay driven by $ G $-Brownian motion ($ G $-SCSND, in short). Combining with inequality technique, $ k $th vertex-Lyapunov functions and graph-theory, we establish the weighted EISS for $ G $-SCSND. An application to the EISS for a class of stochastic coupled oscillators networks with control inputs driven by $ G $-Brownian motion and an example are provided to illustrate the effectiveness of the obtained theory.
References:
[1] |
L. Denis, M. Hu and S. Peng,
Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion pathes, Potential Anal., 34 (2011), 139-161.
doi: 10.1007/s11118-010-9185-x. |
[2] |
F. Gao,
Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl., 119 (2009), 3356-3382.
doi: 10.1016/j.spa.2009.05.010. |
[3] |
S. Gao and B. Ying,
On input-to-state stability for stochastic coupled control systems on networks, Appl. Math. Comput., 262 (2015), 90-101.
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[4] |
F. Hu, Z. Chen and P. Wu,
A general strong law of large numbers for non-additive probabilities and its applications, Statistics, 50 (2016), 733-749.
doi: 10.1080/02331888.2016.1143473. |
[5] |
F. Hu, Z. Chen and D. Zhang,
How big are the increments of G-Brownian motion?, Sci. China Math., 57 (2014), 1687-1700.
doi: 10.1007/s11425-014-4816-0. |
[6] |
F. Hu and Z. Chen,
General laws of large numbers under sublinear expectations, Comm. Statist. Theory Methods, 45 (2016), 4215-4229.
doi: 10.1080/03610926.2014.917677. |
[7] |
M. Hu and S. Peng,
On the representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 25 (2009), 539-546.
doi: 10.1007/s10255-008-8831-1. |
[8] |
X. Li, X. Lin and Y. Lin,
Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion, J. Math. Anal. Appl., 439 (2016), 235-255.
doi: 10.1016/j.jmaa.2016.02.042. |
[9] |
W. Li, H. Su and K. Wang,
Global stability analysis for stochastic coupled systems on networks, Automatica J. IFAC, 47 (2011), 215-220.
doi: 10.1016/j.automatica.2010.10.041. |
[10] |
W. Li, H. Yang, L. Wen and K. Wang,
Global exponential stability for coupled retarded systems on networks: A graph-theoretic approach, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1651-1660.
doi: 10.1016/j.cnsns.2013.09.039. |
[11] |
X. Lou and Q. Ye, Input-to-state stability of stochastic memristive neutral networks with time-varying delay, Math. Probl. Eng., 2015 (2015), Art. ID 140857, 8 pp.
doi: 10.1155/2015/140857. |
[12] |
S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, in: Abel Symp., vol. 2, Springer, Berlin, 2007,541-567.
doi: 10.1007/978-3-540-70847-6_25. |
[13] |
S. Peng,
Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 118 (2008), 2223-2253.
doi: 10.1016/j.spa.2007.10.015. |
[14] |
S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, arXiv: 1002.4546v1 Google Scholar |
[15] |
S. Peng, Theory, methods and meaning of nonlinear expectation theory (in Chinese), Sci Sin Math., 47 (2017), 1223-1254. Google Scholar |
[16] |
Y. Ren, X. Jia and L. Hu,
Exponential stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion, Discrete Conti. Dyn. Syst. Ser-B., 20 (2017), 2157-2169.
doi: 10.3934/dcdsb.2015.20.2157. |
[17] |
Y. Ren, X. Jia and R. Sakthivel,
The p-th moment stability of solution to impulsive stochastic differential equations driven by G-Brownian motion, Appl. Anal., 96 (2017), 988-1003.
doi: 10.1080/00036811.2016.1169529. |
[18] |
Y. Ren and W. Yin, Asymptotical boundedness for stochastic coupled systems on networks with time-varying delay driven by G-Brownian motion, Internat. J. Control.. Google Scholar |
[19] |
Y. Song, W. Sun and F. Jiang, Mean-square exponential input-to-state stability for neutral stochastic neural networks with mixed delays, Neurcomputing, 205 (2016), 195-203. Google Scholar |
[20] |
X. Wu, S. Peng, Y. Tang and W. Zhang,
Input-to-state stability of nonlinear stochastic time-varying systems with impulsive effects, Internat. J. Robust Nonlinear Control, 27 (2017), 1792-1809.
doi: 10.1002/rnc.3637. |
[21] |
C. Zhang and T. Chen,
Exponential stability of stochastic complex networks with multi-weights based on graph theory, Phys. A, 496 (2018), 602-611.
doi: 10.1016/j.physa.2017.12.132. |
[22] |
C. Zhang, W. Li and K. Wang,
Graph-theoretic method on exponential synchronization of stochastic coupled networks with markovian switching, Nonlinear Anal. Hybrid Syst., 15 (2015), 37-51.
doi: 10.1016/j.nahs.2014.07.003. |
[23] |
C. Zhang, W. Li and K. Wang,
Graph theory-based approach for stability analysis of stochastic coupled systems with Lévy noise on networks, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 1698-1709.
doi: 10.1109/TNNLS.2014.2352217. |
[24] |
W. Zhou, L. Teng and D. Xu, Mean-square exponentially input-to-state stability of stochastic Cohen-Grossberg neural networks with time-varying delays, Neurocomputing, 153 (2015), 54-61. Google Scholar |
[25] |
Q. Zhu and J. Cao, Mean-square exponential input-to-state stability of stochastic delayed neutral networks, Neurcomputing, 131 (2014), 157-163. Google Scholar |
[26] |
Q. Zhu, J. Cao and R. Rakkiyappan,
Exponential input-to-state stability of stochastic Cohen-Grossberg neural networks with mixed delays, Nonlinear Dynam., 79 (2015), 1085-1098.
doi: 10.1007/s11071-014-1725-2. |
show all references
References:
[1] |
L. Denis, M. Hu and S. Peng,
Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion pathes, Potential Anal., 34 (2011), 139-161.
doi: 10.1007/s11118-010-9185-x. |
[2] |
F. Gao,
Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl., 119 (2009), 3356-3382.
doi: 10.1016/j.spa.2009.05.010. |
[3] |
S. Gao and B. Ying,
On input-to-state stability for stochastic coupled control systems on networks, Appl. Math. Comput., 262 (2015), 90-101.
doi: 10.1016/j.amc.2015.04.007. |
[4] |
F. Hu, Z. Chen and P. Wu,
A general strong law of large numbers for non-additive probabilities and its applications, Statistics, 50 (2016), 733-749.
doi: 10.1080/02331888.2016.1143473. |
[5] |
F. Hu, Z. Chen and D. Zhang,
How big are the increments of G-Brownian motion?, Sci. China Math., 57 (2014), 1687-1700.
doi: 10.1007/s11425-014-4816-0. |
[6] |
F. Hu and Z. Chen,
General laws of large numbers under sublinear expectations, Comm. Statist. Theory Methods, 45 (2016), 4215-4229.
doi: 10.1080/03610926.2014.917677. |
[7] |
M. Hu and S. Peng,
On the representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 25 (2009), 539-546.
doi: 10.1007/s10255-008-8831-1. |
[8] |
X. Li, X. Lin and Y. Lin,
Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion, J. Math. Anal. Appl., 439 (2016), 235-255.
doi: 10.1016/j.jmaa.2016.02.042. |
[9] |
W. Li, H. Su and K. Wang,
Global stability analysis for stochastic coupled systems on networks, Automatica J. IFAC, 47 (2011), 215-220.
doi: 10.1016/j.automatica.2010.10.041. |
[10] |
W. Li, H. Yang, L. Wen and K. Wang,
Global exponential stability for coupled retarded systems on networks: A graph-theoretic approach, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1651-1660.
doi: 10.1016/j.cnsns.2013.09.039. |
[11] |
X. Lou and Q. Ye, Input-to-state stability of stochastic memristive neutral networks with time-varying delay, Math. Probl. Eng., 2015 (2015), Art. ID 140857, 8 pp.
doi: 10.1155/2015/140857. |
[12] |
S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, in: Abel Symp., vol. 2, Springer, Berlin, 2007,541-567.
doi: 10.1007/978-3-540-70847-6_25. |
[13] |
S. Peng,
Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 118 (2008), 2223-2253.
doi: 10.1016/j.spa.2007.10.015. |
[14] |
S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, arXiv: 1002.4546v1 Google Scholar |
[15] |
S. Peng, Theory, methods and meaning of nonlinear expectation theory (in Chinese), Sci Sin Math., 47 (2017), 1223-1254. Google Scholar |
[16] |
Y. Ren, X. Jia and L. Hu,
Exponential stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion, Discrete Conti. Dyn. Syst. Ser-B., 20 (2017), 2157-2169.
doi: 10.3934/dcdsb.2015.20.2157. |
[17] |
Y. Ren, X. Jia and R. Sakthivel,
The p-th moment stability of solution to impulsive stochastic differential equations driven by G-Brownian motion, Appl. Anal., 96 (2017), 988-1003.
doi: 10.1080/00036811.2016.1169529. |
[18] |
Y. Ren and W. Yin, Asymptotical boundedness for stochastic coupled systems on networks with time-varying delay driven by G-Brownian motion, Internat. J. Control.. Google Scholar |
[19] |
Y. Song, W. Sun and F. Jiang, Mean-square exponential input-to-state stability for neutral stochastic neural networks with mixed delays, Neurcomputing, 205 (2016), 195-203. Google Scholar |
[20] |
X. Wu, S. Peng, Y. Tang and W. Zhang,
Input-to-state stability of nonlinear stochastic time-varying systems with impulsive effects, Internat. J. Robust Nonlinear Control, 27 (2017), 1792-1809.
doi: 10.1002/rnc.3637. |
[21] |
C. Zhang and T. Chen,
Exponential stability of stochastic complex networks with multi-weights based on graph theory, Phys. A, 496 (2018), 602-611.
doi: 10.1016/j.physa.2017.12.132. |
[22] |
C. Zhang, W. Li and K. Wang,
Graph-theoretic method on exponential synchronization of stochastic coupled networks with markovian switching, Nonlinear Anal. Hybrid Syst., 15 (2015), 37-51.
doi: 10.1016/j.nahs.2014.07.003. |
[23] |
C. Zhang, W. Li and K. Wang,
Graph theory-based approach for stability analysis of stochastic coupled systems with Lévy noise on networks, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 1698-1709.
doi: 10.1109/TNNLS.2014.2352217. |
[24] |
W. Zhou, L. Teng and D. Xu, Mean-square exponentially input-to-state stability of stochastic Cohen-Grossberg neural networks with time-varying delays, Neurocomputing, 153 (2015), 54-61. Google Scholar |
[25] |
Q. Zhu and J. Cao, Mean-square exponential input-to-state stability of stochastic delayed neutral networks, Neurcomputing, 131 (2014), 157-163. Google Scholar |
[26] |
Q. Zhu, J. Cao and R. Rakkiyappan,
Exponential input-to-state stability of stochastic Cohen-Grossberg neural networks with mixed delays, Nonlinear Dynam., 79 (2015), 1085-1098.
doi: 10.1007/s11071-014-1725-2. |
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