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July  2019, 24(7): 3379-3393. doi: 10.3934/dcdsb.2018325

Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion

Yong Ren 1,2,, , Huijin Yang 2, and  Wensheng Yin 3,

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

* Corresponding author: Yong Ren, Correspondence to Department of Mathematics, Anhui Normal University, Wuhu 241000, China

Received  April 2018 Revised  July 2018 Published  July 2019 Early access  January 2019

Fund Project: This work is supported by the National Natural Science Foundation of China (11871076).

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.

Keywords: Input to state stability, stochastic coupled system, G-Brownian motion, graph-theory.
Mathematics Subject Classification: Primary: 34A37, 60H10; Secondary: 34D10.
Citation: Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325
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

[15]

S. Peng, Theory, methods and meaning of nonlinear expectation theory (in Chinese), Sci Sin Math., 47 (2017), 1223-1254. 

[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..

[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. 

[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. 

[25]

Q. Zhu and J. Cao, Mean-square exponential input-to-state stability of stochastic delayed neutral networks, Neurcomputing, 131 (2014), 157-163. 

[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

[15]

S. Peng, Theory, methods and meaning of nonlinear expectation theory (in Chinese), Sci Sin Math., 47 (2017), 1223-1254. 

[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..

[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. 

[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. 

[25]

Q. Zhu and J. Cao, Mean-square exponential input-to-state stability of stochastic delayed neutral networks, Neurcomputing, 131 (2014), 157-163. 

[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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