# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021072
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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

* Corresponding author: Jinde Cao

Received  February 2020 Revised  January 2021 Early access March 2021

Fund Project: This work was jointly supported by the Key Project of National Science Foundation of China under Grant No. 61833005, the National Natural Science Foundation of China under grant No. 12001100 and the Fundamental Research Funds for the Central Universities under Grant No. 3207012006C2

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.

Citation: Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $G$-expectation framework. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021072
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, MA, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] J. A. Scheinkman and B. LeBaron, Nonlinear dynamics and stock returns, Journal of Business, 62 (1989), 311-337.  doi: 10.1086/296465.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, MA, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] J. A. Scheinkman and B. LeBaron, Nonlinear dynamics and stock returns, Journal of Business, 62 (1989), 311-337.  doi: 10.1086/296465.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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