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Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework
1. | School of Mathematics, Southeast University, Nanjing 211189, China |
2. | Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea |
In this paper we establish a comparison approach to study stabilization of stochastic differential equations driven by $ G $-Brownian motion with delayed ($ G $-SDDEs for short) feedback control. This theory also extends to a general range of moment order and brings more choices of $ p $. Finally, a simple example is proposed to demonstrate the applications of our theory.
References:
[1] |
J. Cao, H. X. Li and D. W. C. Ho,
Synchronization criteria of Lur'e systems with time delay feedback control, Chaos Solitons Fractals, 23 (2005), 1285-1298.
doi: 10.1016/S0960-0779(04)00380-7. |
[2] |
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. |
[3] |
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. |
[4] |
F. Faizullah, Existence of solutions for $G$-SFDEs with Cauchy-Maruyama approximation scheme, Abstr. Appl. Anal., (2014), Art. ID 809431, 8 pp.
doi: 10.1155/2014/809431. |
[5] |
F. Faizullah,
Existence results and moment estimates for NSFDEs driven by $G$-Brownian motion, J. Comput. Theor. Nanosci., 13 (2016), 4679-4686.
doi: 10.1166/jctn.2016.5336. |
[6] |
F. Faizullah,
A note on $p$th moment estimates for stochastic functional differential equations in the framework of $G$-Brownian motion, Iran. J. Sci. Technol. Trans. A Sci., 41 (2017), 1131-1138.
doi: 10.1007/s40995-016-0067-y. |
[7] |
F. Faizullah, On boundedness and convergence of solutions for neutral stochastic functional differential equations driven by $G$-Brownian motion, Adv. Difference Equ., (2019), Paper No. 289, 16 pp.
doi: 10.1186/s13662-019-2218-x. |
[8] |
F. Faizullah, M. Bux, M. A. Rana and G. ur Rahman, Existence and stability of solutions to non-linear neutral stochastic functional differential equations in the framework of $G$-Brownian motion, Adv. Difference Equ., (2017), Paper No. 350, 14 pp.
doi: 10.1186/s13662-017-1400-2. |
[9] |
F. Faizullah, Q. Zhu and R. Ullah,
The existence-uniqueness and exponential estimate of solutions for SFDEs driven by $G$-Brownian motion, Math. Methods Appl. Sci., 44 (2021), 1639-1650.
doi: 10.1002/mma.6867. |
[10] |
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. |
[11] |
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. |
[12] |
X. He and S. Han and J. Tao, Averaging principle for SDEs of neutral type driven by $G$-Brownian motion, Stoch. Dyn., 19 (2019), 1950004, 22 pp.
doi: 10.1142/S0219493719500047. |
[13] |
M. Hu, X. Ji and G. Liu,
On the strong Markov property for stochastic differential equations driven by $G$-Brownian motion, Stochastic Process. Appl., 131 (2021), 417-453.
doi: 10.1016/j.spa.2020.09.015. |
[14] |
M. Hu, S. Ji, S. Peng and Y. Song,
Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by $G$-Brownian motion, Stochastic Process. Appl., 124 (2014), 1170-1195.
doi: 10.1016/j.spa.2013.10.009. |
[15] |
M. Hu, H. Li, F. Wang and G. Zheng, Invariant and ergodic nonlinear expectations for $G$-diffusion processes, Electron. Commun. Probab., 20 (2015), no. 30, 15 pp.
doi: 10.1214/ECP.v20-3886. |
[16] |
J. Hu, W. Liu, F. Deng and X. Mao,
Advances in stabilisation of hybrid stochastic differential equations by delay feedback control, SIAM J. Control Optim., 58 (2020), 735-754.
doi: 10.1137/19M1270240. |
[17] |
M. Hu and F. Wang, Ergodic BSDEs driven by $G$-Brownian motion and applications, Stoch. Dyn., 18 (2018), 1850050, 35 pp.
doi: 10.1142/S0219493718500508. |
[18] |
M. Hu and F. Wang,
Stochastic optimal control problem with infinite horizon driven by $G$-Brownian motion, ESAIM Control Optim. Calc. Var., 24 (2018), 873-899.
doi: 10.1051/cocv/2017044. |
[19] |
X. Huang and F. Yang, Comparison theorem for path dependent SDEs driven by $G$-Brownian motion, preprint, arXiv: 2004.02652v2. Google Scholar |
[20] |
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. |
[21] |
Q. Lin,
Some properties of stochastic differential equations driven by the $G$-Brownian motion, Acta Math. Sin. (Engl. Ser.), 29 (2013), 923-942.
doi: 10.1007/s10114-013-0701-y. |
[22] |
P. Luo and F. Wang,
Viability for stochastic differential equations driven by $G$-Brownian motion, J. Theoret. Probab., 32 (2019), 395-416.
doi: 10.1007/s10959-017-0791-z. |
[23] |
X. Mao, J. Lam and L. Huang,
Stabilization of hybrid stochastic differential equations by delay feedback control, Systems Control Lett., 57 (2008), 927-935.
doi: 10.1016/j.sysconle.2008.05.002. |
[24] |
X. Mao,
Almost sure exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 53 (2015), 370-389.
doi: 10.1137/140966198. |
[25] |
S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, MA, 1984. |
[26] |
S.-E. A. Mohammed and M. K. R. Scheutzow,
Lyapunov exponents of linear stochastic functional differential equations. Part Ⅱ. Examples and case studies, Ann. Probab., 25 (1997), 1210-1240.
doi: 10.1214/aop/1024404511. |
[27] |
S. Peng, $G$-Brownian motion and dynamic risk measure under volatility uncertainty, preprint, arXiv: 0711.2834v1. Google Scholar |
[28] |
S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, Abel Symp. 2, Springer, Berlin, (2007), 541–567.
doi: 10.1007/978-3-540-70847-6_25. |
[29] |
S. Peng, Nonlinear Expectations and Stochastic Calculus Under Uncertainty–With Robust CLT and $G$-Brownian Motion, Springer-Verlag, GmbH Germany, 2019.
doi: 10.1007/978-3-662-59903-7. |
[30] |
K. Pyragas,
Control of chaos via extended delay feedback, Phys. Lett. A, 206 (1995), 323-330.
doi: 10.1016/0375-9601(95)00654-L. |
[31] |
Y. Ren, X. Jia and L. Hu,
Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2157-2169.
|
[32] |
Y. Ren and W. Yin,
Quasi-sure exponential stabilization of nonlinear systems via intermittent $G$-Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5871-5883.
doi: 10.3934/dcdsb.2019110. |
[33] |
Y. Ren, W. Yin and R. Sakthivel,
Stabilization of stochastic differential equations driven by $G$-Brownian motion with feedback control based on discrete-time state observation, Automatica J. IFAC, 95 (2018), 146-151.
doi: 10.1016/j.automatica.2018.05.039. |
[34] |
Y. Ren, W. Yin and D. Zhu,
Stabilisation of SDEs and applications to synchronisation of stochastic neural network driven by $G$-Brownian motion with state-feedback control, Internat. J. Systems Sci., 50 (2019), 273-282.
doi: 10.1080/00207721.2018.1551973. |
[35] |
J. A. Scheinkman and B. LeBaron,
Nonlinear dynamics and stock returns, Journal of Business, 62 (1989), 311-337.
doi: 10.1086/296465. |
[36] |
R. Ullah and F. Faizullah,
On existence and approximate solutions for stochastic differential equations in the framework of $G$-Brownian motion, Eur. Phys. J. Plus, 132 (2017), 435-443.
doi: 10.1140/epjp/i2017-11700-9. |
[37] |
F. Yang,
Harnack inequality and applications for SDEs driven by $G$-Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 627-635.
doi: 10.1007/s10255-020-0957-9. |
[38] |
W. Yin and J. Cao,
Nonlocal stochastic differential equations with time-varying delay driven by $G$-Brownian motion, Math. Methods Appl. Sci., 43 (2020), 600-612.
doi: 10.1002/mma.5912. |
[39] |
W. Yin and Y. Ren,
Asymptotical boundedness and stability for stochastic differential equations with delay driven by $G$-Brownian motion, Appl. Math. Lett., 74 (2017), 121-126.
doi: 10.1016/j.aml.2017.06.001. |
[40] |
D. Zhang and Z. Chen,
Exponential stability for stochastic differential equations driven by $G$-Brownian motion, Appl. Math. Lett., 25 (2012), 1906-1910.
doi: 10.1016/j.aml.2012.02.063. |
show all references
References:
[1] |
J. Cao, H. X. Li and D. W. C. Ho,
Synchronization criteria of Lur'e systems with time delay feedback control, Chaos Solitons Fractals, 23 (2005), 1285-1298.
doi: 10.1016/S0960-0779(04)00380-7. |
[2] |
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. |
[3] |
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. |
[4] |
F. Faizullah, Existence of solutions for $G$-SFDEs with Cauchy-Maruyama approximation scheme, Abstr. Appl. Anal., (2014), Art. ID 809431, 8 pp.
doi: 10.1155/2014/809431. |
[5] |
F. Faizullah,
Existence results and moment estimates for NSFDEs driven by $G$-Brownian motion, J. Comput. Theor. Nanosci., 13 (2016), 4679-4686.
doi: 10.1166/jctn.2016.5336. |
[6] |
F. Faizullah,
A note on $p$th moment estimates for stochastic functional differential equations in the framework of $G$-Brownian motion, Iran. J. Sci. Technol. Trans. A Sci., 41 (2017), 1131-1138.
doi: 10.1007/s40995-016-0067-y. |
[7] |
F. Faizullah, On boundedness and convergence of solutions for neutral stochastic functional differential equations driven by $G$-Brownian motion, Adv. Difference Equ., (2019), Paper No. 289, 16 pp.
doi: 10.1186/s13662-019-2218-x. |
[8] |
F. Faizullah, M. Bux, M. A. Rana and G. ur Rahman, Existence and stability of solutions to non-linear neutral stochastic functional differential equations in the framework of $G$-Brownian motion, Adv. Difference Equ., (2017), Paper No. 350, 14 pp.
doi: 10.1186/s13662-017-1400-2. |
[9] |
F. Faizullah, Q. Zhu and R. Ullah,
The existence-uniqueness and exponential estimate of solutions for SFDEs driven by $G$-Brownian motion, Math. Methods Appl. Sci., 44 (2021), 1639-1650.
doi: 10.1002/mma.6867. |
[10] |
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. |
[11] |
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. |
[12] |
X. He and S. Han and J. Tao, Averaging principle for SDEs of neutral type driven by $G$-Brownian motion, Stoch. Dyn., 19 (2019), 1950004, 22 pp.
doi: 10.1142/S0219493719500047. |
[13] |
M. Hu, X. Ji and G. Liu,
On the strong Markov property for stochastic differential equations driven by $G$-Brownian motion, Stochastic Process. Appl., 131 (2021), 417-453.
doi: 10.1016/j.spa.2020.09.015. |
[14] |
M. Hu, S. Ji, S. Peng and Y. Song,
Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by $G$-Brownian motion, Stochastic Process. Appl., 124 (2014), 1170-1195.
doi: 10.1016/j.spa.2013.10.009. |
[15] |
M. Hu, H. Li, F. Wang and G. Zheng, Invariant and ergodic nonlinear expectations for $G$-diffusion processes, Electron. Commun. Probab., 20 (2015), no. 30, 15 pp.
doi: 10.1214/ECP.v20-3886. |
[16] |
J. Hu, W. Liu, F. Deng and X. Mao,
Advances in stabilisation of hybrid stochastic differential equations by delay feedback control, SIAM J. Control Optim., 58 (2020), 735-754.
doi: 10.1137/19M1270240. |
[17] |
M. Hu and F. Wang, Ergodic BSDEs driven by $G$-Brownian motion and applications, Stoch. Dyn., 18 (2018), 1850050, 35 pp.
doi: 10.1142/S0219493718500508. |
[18] |
M. Hu and F. Wang,
Stochastic optimal control problem with infinite horizon driven by $G$-Brownian motion, ESAIM Control Optim. Calc. Var., 24 (2018), 873-899.
doi: 10.1051/cocv/2017044. |
[19] |
X. Huang and F. Yang, Comparison theorem for path dependent SDEs driven by $G$-Brownian motion, preprint, arXiv: 2004.02652v2. Google Scholar |
[20] |
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. |
[21] |
Q. Lin,
Some properties of stochastic differential equations driven by the $G$-Brownian motion, Acta Math. Sin. (Engl. Ser.), 29 (2013), 923-942.
doi: 10.1007/s10114-013-0701-y. |
[22] |
P. Luo and F. Wang,
Viability for stochastic differential equations driven by $G$-Brownian motion, J. Theoret. Probab., 32 (2019), 395-416.
doi: 10.1007/s10959-017-0791-z. |
[23] |
X. Mao, J. Lam and L. Huang,
Stabilization of hybrid stochastic differential equations by delay feedback control, Systems Control Lett., 57 (2008), 927-935.
doi: 10.1016/j.sysconle.2008.05.002. |
[24] |
X. Mao,
Almost sure exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 53 (2015), 370-389.
doi: 10.1137/140966198. |
[25] |
S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, MA, 1984. |
[26] |
S.-E. A. Mohammed and M. K. R. Scheutzow,
Lyapunov exponents of linear stochastic functional differential equations. Part Ⅱ. Examples and case studies, Ann. Probab., 25 (1997), 1210-1240.
doi: 10.1214/aop/1024404511. |
[27] |
S. Peng, $G$-Brownian motion and dynamic risk measure under volatility uncertainty, preprint, arXiv: 0711.2834v1. Google Scholar |
[28] |
S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, Abel Symp. 2, Springer, Berlin, (2007), 541–567.
doi: 10.1007/978-3-540-70847-6_25. |
[29] |
S. Peng, Nonlinear Expectations and Stochastic Calculus Under Uncertainty–With Robust CLT and $G$-Brownian Motion, Springer-Verlag, GmbH Germany, 2019.
doi: 10.1007/978-3-662-59903-7. |
[30] |
K. Pyragas,
Control of chaos via extended delay feedback, Phys. Lett. A, 206 (1995), 323-330.
doi: 10.1016/0375-9601(95)00654-L. |
[31] |
Y. Ren, X. Jia and L. Hu,
Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2157-2169.
|
[32] |
Y. Ren and W. Yin,
Quasi-sure exponential stabilization of nonlinear systems via intermittent $G$-Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5871-5883.
doi: 10.3934/dcdsb.2019110. |
[33] |
Y. Ren, W. Yin and R. Sakthivel,
Stabilization of stochastic differential equations driven by $G$-Brownian motion with feedback control based on discrete-time state observation, Automatica J. IFAC, 95 (2018), 146-151.
doi: 10.1016/j.automatica.2018.05.039. |
[34] |
Y. Ren, W. Yin and D. Zhu,
Stabilisation of SDEs and applications to synchronisation of stochastic neural network driven by $G$-Brownian motion with state-feedback control, Internat. J. Systems Sci., 50 (2019), 273-282.
doi: 10.1080/00207721.2018.1551973. |
[35] |
J. A. Scheinkman and B. LeBaron,
Nonlinear dynamics and stock returns, Journal of Business, 62 (1989), 311-337.
doi: 10.1086/296465. |
[36] |
R. Ullah and F. Faizullah,
On existence and approximate solutions for stochastic differential equations in the framework of $G$-Brownian motion, Eur. Phys. J. Plus, 132 (2017), 435-443.
doi: 10.1140/epjp/i2017-11700-9. |
[37] |
F. Yang,
Harnack inequality and applications for SDEs driven by $G$-Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 627-635.
doi: 10.1007/s10255-020-0957-9. |
[38] |
W. Yin and J. Cao,
Nonlocal stochastic differential equations with time-varying delay driven by $G$-Brownian motion, Math. Methods Appl. Sci., 43 (2020), 600-612.
doi: 10.1002/mma.5912. |
[39] |
W. Yin and Y. Ren,
Asymptotical boundedness and stability for stochastic differential equations with delay driven by $G$-Brownian motion, Appl. Math. Lett., 74 (2017), 121-126.
doi: 10.1016/j.aml.2017.06.001. |
[40] |
D. Zhang and Z. Chen,
Exponential stability for stochastic differential equations driven by $G$-Brownian motion, Appl. Math. Lett., 25 (2012), 1906-1910.
doi: 10.1016/j.aml.2012.02.063. |
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