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November  2019, 24(11): 5871-5883. doi: 10.3934/dcdsb.2019110

Quasi sure exponential stabilization of nonlinear systems via intermittent $ G $-Brownian motion

Yong Ren 1,, and  Wensheng Yin 2,

1. 

Department of Mathematics, Anhui Normal University, Wuhu 241000, China
2. 

School of Mathematics, Southeast University, Nanjing 211189, China

* Corresponding author

Received  September 2018 Published  November 2019 Early access  June 2019

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

This paper focuses on the quasi sure exponential stabilization of nonlinear systems. By virtue of exponential martingale inequality under $ G $-framework and intermittent $ G $-Brownian motion (in short, $ G $-ISSs), we establish the sufficient conditions to guarantee quasi surely exponential stability. The efficiency of the proposed results is illustrated by the memristor-based Chua's oscillator.

Keywords: Quasi surely exponential stabilization, nonlinear systems, G-Brownian motion.
Mathematics Subject Classification: Primary: 34A37, 60H10; Secondary: 34D10.
Citation: Yong Ren, Wensheng Yin. Quasi sure exponential stabilization of nonlinear systems via intermittent $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5871-5883. doi: 10.3934/dcdsb.2019110
References:
[1]

Z. Chen, P. Wu and B. Li, A strong law of large numbers for non-additive probabilities, Internat. J. Approx. Reason., 54 (2013), 365-377.  doi: 10.1016/j.ijar.2012.06.002.  Google Scholar

[2]

P. Cheng, F. Deng and F. Yao, Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects, Nonlinear Anal. Hybrid Syst., 30 (2018), 106-117.  doi: 10.1016/j.nahs.2018.05.003.  Google Scholar

[3]

F. Deng, Q. Luo and X. Mao, Stochastic stabilization of hybrid differential equations, Automatica, 48 (2012), 2321-2328.  doi: 10.1016/j.automatica.2012.06.044.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

Q. Guo, X. Mao and R. Yue, Almost sure exponential stability of stochastic differential delay equations, SIAM J. Control Optim., 54 (2016), 1919-1933.  doi: 10.1137/15M1019465.  Google Scholar

[7]

B. Guo, Y. Wu, Y. Xiao and C. Zhang, Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control, Appl. Math. Comput., 331 (2018), 341-357.  doi: 10.1016/j.amc.2018.03.020.  Google Scholar

[8]

F. Hu, The modulus of continuity theorem for G-Brownian motion, Comm. Statist. Theory Methods, 46 (2017), 3586-3598.  doi: 10.1080/03610926.2015.1066816.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

F. Hu and D. Zhang, Central limit theorem for capacities, C. R. Math. Acad. Sci. Paris, 348 (2010), 1111-1114.  doi: 10.1016/j.crma.2010.07.026.  Google Scholar

[13]

L. Hu and X. Mao, Almost sure exponential stabilization of stochastic systems by state feedback control, Automatica J. IFAC., 44 (2008), 465-471.  doi: 10.1016/j.automatica.2007.05.027.  Google Scholar

[14]

C. Hu, J. Yu, H. Jiang and Z. Teng, Exponential stabilization and synchronization of neural networks with time-varying delays via periodically intermittent control, Nonlinearity, 23 (2010), 2369-2391.  doi: 10.1088/0951-7715/23/10/002.  Google Scholar

[15]

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.  Google Scholar

[16]

X. Mao, stochastic stabilization and destabilization, Syst. Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[17]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. ⅹⅷ+422 pp. ISBN: 978-1-904275-34-3. doi: 10.1533/9780857099402.  Google Scholar

[18]

X. Mao, J. Lam and L. Huang, Stabilization of hybrid stochastic differential equations by delay feedback control, System Control Lett., 57 (2008), 927-935.  doi: 10.1016/j.sysconle.2008.05.002.  Google Scholar

[19]

X. Mao, G. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar

[20]

X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Tran. Autom. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.  Google Scholar

[21]

S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, in: Abel Symp., Springer, Berlin, 2 (2007), 541–567. doi: 10.1007/978-3-540-70847-6_25.  Google Scholar

[22]

Y. Ren, W. Yin and D. Zhu, Exponential stability of SDEs driven by G-Brownian motion with delayed impulsive effects: average impulsive interval approach, Discrete Conti. Dyn. Syst. Ser–B., 23 (2018), 3347-3360.  doi: 10.3934/dcdsb.2018248.  Google Scholar

[23]

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.  Google Scholar

[24]

M. Song and X. Mao, Almost sure exponential stability of hybrid stochastic functional differential equations, J. Math. Anal. Appl., 458 (2018), 1390-1408.  doi: 10.1016/j.jmaa.2017.10.042.  Google Scholar

[25]

F. Wu, X. Mao and S. Hu, Stochastic suppression and stabilization of functional differential equations, System Control Lett., 59 (2010), 745-753.  doi: 10.1016/j.sysconle.2010.08.011.  Google Scholar

[26]

F. Wu, X. Mao and P. E. Kloeden, Almost sure exponential stability of the Euler-Maruyama approximations for stochastic functional differential equations, Random Oper. Stoch. Equ., 19 (2011), 165-186.  doi: 10.1515/ROSE.2011.010.  Google Scholar

[27]

S. Yang, C. Li and T. Huang, Exponential stabilization and synchronization for fuzzy model of memristive neural networks by periodically intermittent control, Neural Networks, 75 (2016), 162-172.  doi: 10.1016/j.neunet.2015.12.003.  Google Scholar

[28]

D. Zhang and Z. Chen, Exponential stability for stochastic differential equations driven by G-Brownian motion, Appl. Math. Letter., 25 (2012), 1906-1910.  doi: 10.1016/j.aml.2012.02.063.  Google Scholar

[29]

B. Zhang, F. Deng, S. Peng and S. Xie, Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system, J. Franklin Inst., 355 (2018), 3829-3852.  doi: 10.1016/j.jfranklin.2017.12.033.  Google Scholar

[30]

S. Zhu, K. Sun, S. Zhou and Y. Shi, Stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial, J. Franklin Inst., 354 (2017), 6550-6566.  doi: 10.1016/j.jfranklin.2017.08.007.  Google Scholar

show all references

References:
[1]

Z. Chen, P. Wu and B. Li, A strong law of large numbers for non-additive probabilities, Internat. J. Approx. Reason., 54 (2013), 365-377.  doi: 10.1016/j.ijar.2012.06.002.  Google Scholar

[2]

P. Cheng, F. Deng and F. Yao, Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects, Nonlinear Anal. Hybrid Syst., 30 (2018), 106-117.  doi: 10.1016/j.nahs.2018.05.003.  Google Scholar

[3]

F. Deng, Q. Luo and X. Mao, Stochastic stabilization of hybrid differential equations, Automatica, 48 (2012), 2321-2328.  doi: 10.1016/j.automatica.2012.06.044.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

Q. Guo, X. Mao and R. Yue, Almost sure exponential stability of stochastic differential delay equations, SIAM J. Control Optim., 54 (2016), 1919-1933.  doi: 10.1137/15M1019465.  Google Scholar

[7]

B. Guo, Y. Wu, Y. Xiao and C. Zhang, Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control, Appl. Math. Comput., 331 (2018), 341-357.  doi: 10.1016/j.amc.2018.03.020.  Google Scholar

[8]

F. Hu, The modulus of continuity theorem for G-Brownian motion, Comm. Statist. Theory Methods, 46 (2017), 3586-3598.  doi: 10.1080/03610926.2015.1066816.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

F. Hu and D. Zhang, Central limit theorem for capacities, C. R. Math. Acad. Sci. Paris, 348 (2010), 1111-1114.  doi: 10.1016/j.crma.2010.07.026.  Google Scholar

[13]

L. Hu and X. Mao, Almost sure exponential stabilization of stochastic systems by state feedback control, Automatica J. IFAC., 44 (2008), 465-471.  doi: 10.1016/j.automatica.2007.05.027.  Google Scholar

[14]

C. Hu, J. Yu, H. Jiang and Z. Teng, Exponential stabilization and synchronization of neural networks with time-varying delays via periodically intermittent control, Nonlinearity, 23 (2010), 2369-2391.  doi: 10.1088/0951-7715/23/10/002.  Google Scholar

[15]

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.  Google Scholar

[16]

X. Mao, stochastic stabilization and destabilization, Syst. Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[17]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. ⅹⅷ+422 pp. ISBN: 978-1-904275-34-3. doi: 10.1533/9780857099402.  Google Scholar

[18]

X. Mao, J. Lam and L. Huang, Stabilization of hybrid stochastic differential equations by delay feedback control, System Control Lett., 57 (2008), 927-935.  doi: 10.1016/j.sysconle.2008.05.002.  Google Scholar

[19]

X. Mao, G. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar

[20]

X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Tran. Autom. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.  Google Scholar

[21]

S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, in: Abel Symp., Springer, Berlin, 2 (2007), 541–567. doi: 10.1007/978-3-540-70847-6_25.  Google Scholar

[22]

Y. Ren, W. Yin and D. Zhu, Exponential stability of SDEs driven by G-Brownian motion with delayed impulsive effects: average impulsive interval approach, Discrete Conti. Dyn. Syst. Ser–B., 23 (2018), 3347-3360.  doi: 10.3934/dcdsb.2018248.  Google Scholar

[23]

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.  Google Scholar

[24]

M. Song and X. Mao, Almost sure exponential stability of hybrid stochastic functional differential equations, J. Math. Anal. Appl., 458 (2018), 1390-1408.  doi: 10.1016/j.jmaa.2017.10.042.  Google Scholar

[25]

F. Wu, X. Mao and S. Hu, Stochastic suppression and stabilization of functional differential equations, System Control Lett., 59 (2010), 745-753.  doi: 10.1016/j.sysconle.2010.08.011.  Google Scholar

[26]

F. Wu, X. Mao and P. E. Kloeden, Almost sure exponential stability of the Euler-Maruyama approximations for stochastic functional differential equations, Random Oper. Stoch. Equ., 19 (2011), 165-186.  doi: 10.1515/ROSE.2011.010.  Google Scholar

[27]

S. Yang, C. Li and T. Huang, Exponential stabilization and synchronization for fuzzy model of memristive neural networks by periodically intermittent control, Neural Networks, 75 (2016), 162-172.  doi: 10.1016/j.neunet.2015.12.003.  Google Scholar

[28]

D. Zhang and Z. Chen, Exponential stability for stochastic differential equations driven by G-Brownian motion, Appl. Math. Letter., 25 (2012), 1906-1910.  doi: 10.1016/j.aml.2012.02.063.  Google Scholar

[29]

B. Zhang, F. Deng, S. Peng and S. Xie, Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system, J. Franklin Inst., 355 (2018), 3829-3852.  doi: 10.1016/j.jfranklin.2017.12.033.  Google Scholar

[30]

S. Zhu, K. Sun, S. Zhou and Y. Shi, Stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial, J. Franklin Inst., 354 (2017), 6550-6566.  doi: 10.1016/j.jfranklin.2017.08.007.  Google Scholar

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