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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 |
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.
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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. |
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