# American Institute of Mathematical Sciences

February  2021, 26(2): 755-774. doi: 10.3934/dcdsb.2020133

## Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control

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

* Corresponding author: Guangjun Shen

Received  November 2019 Revised  January 2020 Published  April 2020

Fund Project: This work has been partially supported by the Top talent project of university discipline (specialty) (gxbjZD03), the Distinguished Young Scholars Foundation of Anhui Province (1608085J06), the National Natural Science Foundation of China (11901005)

The stabilization of stochastic differential equations driven by Brownian motion (G-Brownian motion) with discrete-time feedback controls under Lipschitz conditions has been discussed by several authors. In this paper, we first give the sufficient condition for the mean square exponential instability of stochastic differential equations driven by G-Lévy process with non-Lipschitz coefficients. Second, we design a discrete-time feedback control in the drift part and obtain the mean square exponential stability and quasi-sure exponential stability for the controlled systems. At last, we give an example to verify the obtained theory.

Citation: Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
##### References:
 [1] X.-P. Bai and Y.-Q. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 589-610.  doi: 10.1007/s10255-014-0405-9.  Google Scholar [2] L. Denis, M. S. Hu and S. G. Peng, Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths, Potential Analysis, 34 (2011), 139-161.  doi: 10.1007/s11118-010-9185-x.  Google Scholar [3] C. Fei, W. Y. Fei, X. R. Mao, D. F. Xia and L. T. Yan, Stabilisation of highly nonlinear hybrid systems by feedback control based on discrete-time state observations, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, (2019), http://dx.doi.org/10.1109/TAC.2019.2933604. Google Scholar [4] F. Q. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion, Stochastic Processes and Their Applications, 119 (2009), 3356-3382.  doi: 10.1016/j.spa.2009.05.010.  Google Scholar [5] F. Q. Gao and H. Jiang, Large deviations for stochastic differential equations driven by $G$-Brownian motion, Stochastic Processes and Their Applications, 120 (2010), 2212-2240.  doi: 10.1016/j.spa.2010.06.007.  Google Scholar [6] L. J. Hu and X. R. Mao, Almost sure exponential stabilisation of stochastic systems by state-feedback control, Automatica J. IFAC, 44 (2008), 465-471.  doi: 10.1016/j.automatica.2007.05.027.  Google Scholar [7] M. S. Hu and S. G. Peng, G-Lévy processes under Sublinear Expectations, arXiv: 0911.3533v1. Google Scholar [8] M. L. Li, F. Q. Deng and X. R. Mao, Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps, Science China Information Sciences, 62 (2019), 012204, 15 pp. doi: 10.1007/s11432-017-9302-9.  Google Scholar [9] X. P. Li and S. G. Peng, Stopping times and related Itô's calculus with $G$-Brownian motion, Stochastic Processes and Their Applications, 121 (2011), 1492-1508.  doi: 10.1016/j.spa.2011.03.009.  Google Scholar [10] X. P. Li, X. Y. Lin and Y. Q. Lin, Lyapunov-type conditions and stochastic differential equations driven by $G$-Brownian motion, Journal of Mathematical Analysis and Applications, 439 (2016), 235-255.  doi: 10.1016/j.jmaa.2016.02.042.  Google Scholar [11] X. Y. Li and X. R. Mao, Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control, Automatica J. IFAC, 112 (2020), 108657, 11 pp. doi: 10.1016/j.automatica.2019.108657.  Google Scholar [12] X. R. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.  Google Scholar [13] X. R. Mao, Stochastic differential equations and their applications (2nd ed.), Elsevier, (2007). Google Scholar [14] X. R. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Trans. Automat. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.  Google Scholar [15] A. Neufeld and M. Nutz, Nonlinear Lévy processes and their characteristics, Transactions of the American Mathematical Society, 369 (2017), 69-95.  doi: 10.1090/tran/6656.  Google Scholar [16] K. Paczka, Itô calculus and jump diffusions for G-Lévy processes, arXiv: 1211.2973v3. Google Scholar [17] K. Paczka, On the properites of Poisson random measures associated with a G-Lévy process, arXiv: 1411.4660v1. Google Scholar [18] K. Paczka, G-martingale representation in the G-Lévy setting, arXiv: 1404.2121v1. Google Scholar [19] S. G. Peng, Nonlinear Expectations and Stochastic Calculus under Uncertainty: With Robust CLT and G-Brownian Motion, Probability Theory and Stochastic Modelling, 95. Springer, Berlin, 2019, arXiv: 1002.4546v1. doi: 10.1007/978-3-662-59903-7.  Google Scholar [20] S. G. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expections, Science in China. Series A, 52 (2009), 1391-1411.  doi: 10.1007/s11425-009-0121-8.  Google Scholar [21] S. G. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, Abel Symp., Springer, Berlin, 2 (2007), 541-567.  doi: 10.1007/978-3-540-70847-6_25.  Google Scholar [22] S. G. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under $G$-expectation, Stochastic Processes and Their Applications, 118 (2008), 2223-2253.  doi: 10.1016/j.spa.2007.10.015.  Google Scholar [23] Q. Qiu, W. Liu, L. Hu and J. Lu, Stabilisation of hybrid stochastic systems under discrete observation and sample delay, Control Theory and Applications, 33 (2016), 1024-1030.   Google Scholar [24] L. Y. Ren, On representation theorem of sublinear expectation related to $G$-Lévy process and paths of $G$-Lévy process, Statistics and Probability Letters, 83 (2013), 1301–1310. doi: 10.1016/j.spl.2013.01.031.  Google Scholar [25] Y. Ren, W. S. 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 [26] Y. Ren, W. S. Yin and D. J. Zhu, Stabilisation of SDEs and applications to synchronisation of stochastic neural network driven by G-Brownian motion with state-feedback control, International Journal of Systems Science, 50 (2019), 273-282.  doi: 10.1080/00207721.2018.1551973.  Google Scholar [27] Y. Ren, X. J. Jia and R. Sakthivel, The pth moment stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion, Applicable Analysis, 96 (2017), 988-1003.  doi: 10.1080/00036811.2016.1169529.  Google Scholar [28] J. H. Shao, Stabilization of regime-switching processes by feedback control based on discrete time state observations, SIAM Journal on Control and Optimization, 55 (2017), 724-740.  doi: 10.1137/16M1066336.  Google Scholar [29] G. F. Song, Z. Y. Lu, B.-C. Zheng and X. R. Mao, Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state, Science China Information Sciences, 61 (2018), 070213, 16 pp. doi: 10.1007/s11432-017-9297-1.  Google Scholar [30] H. M. Soner, N. Touzi and J. F. Zhang, Martingale representation theorem for the $G$-expectation, Stochastic Processes and Their Applications, 121 (2011), 265-287.  doi: 10.1016/j.spa.2010.10.006.  Google Scholar [31] B. J. Wang and M. X. Yuan, Existence of solution for stochastic differential equations driven by $G$-Lévy processes with discontinuous coefficients, Advances in Difference Equations, 2017 (2017), 13 pp. doi: 10.1186/s13662-017-1242-y.  Google Scholar [32] B. J. Wang and H. J. Gao, Exponential stability of solutions to stochastic differential equations driven by $G$-Lévy Process, Applied Mathematics and Optimization, (2019). doi: 10.1007/s00245-019-09583-0.  Google Scholar [33] S. R. You, W. Liu, J. Q. Liu, X. R. Mao and Q. W. Qiu, Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM Journal on Control and Optimization, 53 (2015), 905-925.  doi: 10.1137/140985779.  Google Scholar

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##### References:
 [1] X.-P. Bai and Y.-Q. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 589-610.  doi: 10.1007/s10255-014-0405-9.  Google Scholar [2] L. Denis, M. S. Hu and S. G. Peng, Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths, Potential Analysis, 34 (2011), 139-161.  doi: 10.1007/s11118-010-9185-x.  Google Scholar [3] C. Fei, W. Y. Fei, X. R. Mao, D. F. Xia and L. T. Yan, Stabilisation of highly nonlinear hybrid systems by feedback control based on discrete-time state observations, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, (2019), http://dx.doi.org/10.1109/TAC.2019.2933604. Google Scholar [4] F. Q. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion, Stochastic Processes and Their Applications, 119 (2009), 3356-3382.  doi: 10.1016/j.spa.2009.05.010.  Google Scholar [5] F. Q. Gao and H. Jiang, Large deviations for stochastic differential equations driven by $G$-Brownian motion, Stochastic Processes and Their Applications, 120 (2010), 2212-2240.  doi: 10.1016/j.spa.2010.06.007.  Google Scholar [6] L. J. Hu and X. R. Mao, Almost sure exponential stabilisation of stochastic systems by state-feedback control, Automatica J. IFAC, 44 (2008), 465-471.  doi: 10.1016/j.automatica.2007.05.027.  Google Scholar [7] M. S. Hu and S. G. Peng, G-Lévy processes under Sublinear Expectations, arXiv: 0911.3533v1. Google Scholar [8] M. L. Li, F. Q. Deng and X. R. Mao, Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps, Science China Information Sciences, 62 (2019), 012204, 15 pp. doi: 10.1007/s11432-017-9302-9.  Google Scholar [9] X. P. Li and S. G. Peng, Stopping times and related Itô's calculus with $G$-Brownian motion, Stochastic Processes and Their Applications, 121 (2011), 1492-1508.  doi: 10.1016/j.spa.2011.03.009.  Google Scholar [10] X. P. Li, X. Y. Lin and Y. Q. Lin, Lyapunov-type conditions and stochastic differential equations driven by $G$-Brownian motion, Journal of Mathematical Analysis and Applications, 439 (2016), 235-255.  doi: 10.1016/j.jmaa.2016.02.042.  Google Scholar [11] X. Y. Li and X. R. Mao, Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control, Automatica J. IFAC, 112 (2020), 108657, 11 pp. doi: 10.1016/j.automatica.2019.108657.  Google Scholar [12] X. R. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.  Google Scholar [13] X. R. Mao, Stochastic differential equations and their applications (2nd ed.), Elsevier, (2007). Google Scholar [14] X. R. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Trans. Automat. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.  Google Scholar [15] A. Neufeld and M. Nutz, Nonlinear Lévy processes and their characteristics, Transactions of the American Mathematical Society, 369 (2017), 69-95.  doi: 10.1090/tran/6656.  Google Scholar [16] K. Paczka, Itô calculus and jump diffusions for G-Lévy processes, arXiv: 1211.2973v3. Google Scholar [17] K. Paczka, On the properites of Poisson random measures associated with a G-Lévy process, arXiv: 1411.4660v1. Google Scholar [18] K. Paczka, G-martingale representation in the G-Lévy setting, arXiv: 1404.2121v1. Google Scholar [19] S. G. Peng, Nonlinear Expectations and Stochastic Calculus under Uncertainty: With Robust CLT and G-Brownian Motion, Probability Theory and Stochastic Modelling, 95. Springer, Berlin, 2019, arXiv: 1002.4546v1. doi: 10.1007/978-3-662-59903-7.  Google Scholar [20] S. G. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expections, Science in China. Series A, 52 (2009), 1391-1411.  doi: 10.1007/s11425-009-0121-8.  Google Scholar [21] S. G. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type, Stochastic Analysis and Applications, Abel Symp., Springer, Berlin, 2 (2007), 541-567.  doi: 10.1007/978-3-540-70847-6_25.  Google Scholar [22] S. G. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under $G$-expectation, Stochastic Processes and Their Applications, 118 (2008), 2223-2253.  doi: 10.1016/j.spa.2007.10.015.  Google Scholar [23] Q. Qiu, W. Liu, L. Hu and J. Lu, Stabilisation of hybrid stochastic systems under discrete observation and sample delay, Control Theory and Applications, 33 (2016), 1024-1030.   Google Scholar [24] L. Y. Ren, On representation theorem of sublinear expectation related to $G$-Lévy process and paths of $G$-Lévy process, Statistics and Probability Letters, 83 (2013), 1301–1310. doi: 10.1016/j.spl.2013.01.031.  Google Scholar [25] Y. Ren, W. S. 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 [26] Y. Ren, W. S. Yin and D. J. Zhu, Stabilisation of SDEs and applications to synchronisation of stochastic neural network driven by G-Brownian motion with state-feedback control, International Journal of Systems Science, 50 (2019), 273-282.  doi: 10.1080/00207721.2018.1551973.  Google Scholar [27] Y. Ren, X. J. Jia and R. Sakthivel, The pth moment stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion, Applicable Analysis, 96 (2017), 988-1003.  doi: 10.1080/00036811.2016.1169529.  Google Scholar [28] J. H. Shao, Stabilization of regime-switching processes by feedback control based on discrete time state observations, SIAM Journal on Control and Optimization, 55 (2017), 724-740.  doi: 10.1137/16M1066336.  Google Scholar [29] G. F. Song, Z. Y. Lu, B.-C. Zheng and X. R. Mao, Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state, Science China Information Sciences, 61 (2018), 070213, 16 pp. doi: 10.1007/s11432-017-9297-1.  Google Scholar [30] H. M. Soner, N. Touzi and J. F. Zhang, Martingale representation theorem for the $G$-expectation, Stochastic Processes and Their Applications, 121 (2011), 265-287.  doi: 10.1016/j.spa.2010.10.006.  Google Scholar [31] B. J. Wang and M. X. Yuan, Existence of solution for stochastic differential equations driven by $G$-Lévy processes with discontinuous coefficients, Advances in Difference Equations, 2017 (2017), 13 pp. doi: 10.1186/s13662-017-1242-y.  Google Scholar [32] B. J. Wang and H. J. Gao, Exponential stability of solutions to stochastic differential equations driven by $G$-Lévy Process, Applied Mathematics and Optimization, (2019). doi: 10.1007/s00245-019-09583-0.  Google Scholar [33] S. R. You, W. Liu, J. Q. Liu, X. R. Mao and Q. W. Qiu, Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM Journal on Control and Optimization, 53 (2015), 905-925.  doi: 10.1137/140985779.  Google Scholar
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