• Previous Article
    On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium
  • DCDS-B Home
  • This Issue
  • Next Article
    Modelling the effects of contaminated environments in mainland China on seasonal HFMD infections and the potential benefit of a pulse vaccination strategy
November  2019, 24(11): 5871-5883. doi: 10.3934/dcdsb.2019110

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

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.

Citation: Yong Ren, Wensheng Yin. Quasi sure exponential stabilization of nonlinear systems via intermittent $ G $-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5871-5883. doi: 10.3934/dcdsb.2019110
References:
[1]

Z. ChenP. 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.

[2]

P. ChengF. 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.

[3]

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

[4]

L. DenisM. 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.

[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.

[6]

Q. GuoX. 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.

[7]

B. GuoY. WuY. 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.

[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.

[9]

F. HuZ. 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.

[10]

F. HuZ. 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.

[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.

[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.

[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.

[14]

C. HuJ. YuH. 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.

[15]

X. LiX. 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.

[16]

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

[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.

[18]

X. MaoJ. 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.

[19]

X. MaoG. 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.

[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.

[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.

[22]

Y. RenW. 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.

[23]

Y. RenX. 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.

[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.

[25]

F. WuX. 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.

[26]

F. WuX. 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.

[27]

S. YangC. 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.

[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.

[29]

B. ZhangF. DengS. 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.

[30]

S. ZhuK. SunS. 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.

show all references

References:
[1]

Z. ChenP. 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.

[2]

P. ChengF. 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.

[3]

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

[4]

L. DenisM. 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.

[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.

[6]

Q. GuoX. 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.

[7]

B. GuoY. WuY. 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.

[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.

[9]

F. HuZ. 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.

[10]

F. HuZ. 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.

[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.

[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.

[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.

[14]

C. HuJ. YuH. 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.

[15]

X. LiX. 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.

[16]

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

[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.

[18]

X. MaoJ. 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.

[19]

X. MaoG. 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.

[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.

[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.

[22]

Y. RenW. 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.

[23]

Y. RenX. 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.

[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.

[25]

F. WuX. 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.

[26]

F. WuX. 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.

[27]

S. YangC. 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.

[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.

[29]

B. ZhangF. DengS. 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.

[30]

S. ZhuK. SunS. 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.

[1]

Francesca Biagini, Thilo Meyer-Brandis, Bernt Øksendal, Krzysztof Paczka. Optimal control with delayed information flow of systems driven by G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 8-. doi: 10.1186/s41546-018-0033-z

[2]

Zhengyan Lin, Li-Xin Zhang. Convergence to a self-normalized G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 4-. doi: 10.1186/s41546-017-0013-8

[3]

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

[4]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157

[5]

Yong Ren, Wensheng Yin, Dongjin Zhu. Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3347-3360. doi: 10.3934/dcdsb.2018248

[6]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

[7]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197

[8]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257

[9]

Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015

[10]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[11]

Xin Meng, Cunchen Gao, Baoping Jiang, Hamid Reza Karimi. Observer-based SMC for stochastic systems with disturbance driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022027

[12]

Sérgio S. Rodrigues. Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems. Evolution Equations and Control Theory, 2020, 9 (3) : 635-672. doi: 10.3934/eect.2020027

[13]

Fabrice Baudoin, Camille Tardif. Hypocoercive estimates on foliations and velocity spherical Brownian motion. Kinetic and Related Models, 2018, 11 (1) : 1-23. doi: 10.3934/krm.2018001

[14]

Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063

[15]

Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075

[16]

Yaofeng Su. Almost surely invariance principle for non-stationary and random intermittent dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6585-6597. doi: 10.3934/dcds.2019286

[17]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255

[18]

Dingjun Yao, Rongming Wang, Lin Xu. Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission. Journal of Industrial and Management Optimization, 2015, 11 (2) : 461-478. doi: 10.3934/jimo.2015.11.461

[19]

Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3725-3757. doi: 10.3934/dcds.2021014

[20]

Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations and Control Theory, 2022, 11 (3) : 925-937. doi: 10.3934/eect.2021031

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (182)
  • HTML views (284)
  • Cited by (1)

Other articles
by authors

[Back to Top]