# American Institute of Mathematical Sciences

January  2017, 22(1): 209-226. doi: 10.3934/dcdsb.2017011

## pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations

 1 Department of Mathematics, Science of College, China University of Petroleum, Beijing 102249, China 2 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China and College of Biochemical Engineering, Beijing Union University, Beijing 100101, China

* Corresponding author: Yi Zhang

Received  August 2015 Revised  May 2016 Published  December 2016

In this paper, we discuss the $p$th moment exponential stabilization of continuous-time hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. The hybrid stochastic functional differential equations are also known as stochastic functional differential equations with the Markovian switching. We follow Mao's paper to consider the auxiliary system whose control is based on continuous-time state observation. The lemma is provided that if the $p$th moment of the solution $y(t)$ of the auxiliary system decays exponentially then the same with the $p$th moment of the functional $y_t$. With the help of this lemma, the criterion for $p$th moment exponential stability of the primary system is given, and the margin of the duration of the discrete-time state observation is presented. Then the special case like the linear system is considered and the discrete-time feedback control is designed.

Citation: Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011
##### References:
 [1] W. F. Ames and B. G. Pachpatte, Inequalities for Differential and Integral Equations Academic Press, 197 (1997).Google Scholar [2] G. K. Basak, A. Bisi and M. K. Ghosh, Stability of a random diffusion with linear drift, Journal of Mathematical Analysis and Applications, 202 (1996), 604-622. doi: 10.1006/jmaa.1996.0336. Google Scholar [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. Google Scholar [4] Y. Ji and H. J. Chizeck, Controllability, stabilizability, and continuous-time markovian jump linear quadratic control, Automatic Control IEEE Transactions on, 35 (1990), 777-788. doi: 10.1109/9.57016. Google Scholar [5] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Processes and Their Applications, 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2. Google Scholar [6] X. Mao, Stochastic functional differential equations with markovian switching, Funct. Differ. Equ, 6 (1999), 375-396. Google Scholar [7] X. Mao, Asymptotic stability for stochastic differential equations with Markovian switching, Funct. Differ. Equ., 9 (2002), 201-220. Google Scholar [8] X. Mao, Exponential stability of stochastic delay interval systems with markovian switching, Automatic Control IEEE Transactions on, 47 (2002), 1604-1612. doi: 10.1109/TAC.2002.803529. Google Scholar [9] 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. Google Scholar [10] X. Mao, J. Lam and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Systems Control Letters, 57 (2008), 927-935. doi: 10.1016/j.sysconle.2008.05.002. Google Scholar [11] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching London: Imperial College Press, 2006. doi: 10.1142/p473. Google Scholar [12] X. Mao, A. Matasov and A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli, 6 (2000), 73-90. doi: 10.2307/3318634. Google Scholar [13] X. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica, 49 (2013), 3677-3681. doi: 10.1016/j.automatica.2013.09.005. Google Scholar [14] X. Mao, W. Liu, L. Hu, Q. Luo and J. Lu, Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Systems & Control Letters, 73 (2014), 88-95. doi: 10.1016/j.sysconle.2014.08.011. Google Scholar [15] S. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, 99. Pitman (Advanced Publishing Program), Boston, MA, 1984. Google Scholar [16] Z. Yang, X. Mao and C. Yuan, Comparison theorem of one-dimensional stochastic hybrid delay systems, Systems & Control Letters, 57 (2008), 56-63. doi: 10.1016/j.sysconle.2007.06.014. Google Scholar [17] S. You, W. Liu, J. Lu, X. Mao and Q. 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 [18] C. Yuan and X. Mao, Robust stability and controllability of stochastic differential delay equations with markovian switching, Automatica, 40 (2004), 343-354. doi: 10.1016/j.automatica.2003.10.012. Google Scholar [19] H. Xu, K. L. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Trans. on Sys. Man. and Cyber. B, 38 (2008), 1419-1422. Google Scholar [20] H. Xu, Y. Chen and K. L. Teo, Global exponential stability of impulsive discrete-time neural networks with time-varying delays, Applied Mathematics and Computation, 217 (2010), 537-544. doi: 10.1016/j.amc.2010.05.087. Google Scholar [21] Y. Zhang, Y. Zhao, M. Shi, H. Shi and C. Liu, The absolute stability of stochastic control system with Markovian switching, Dynam. Cont. Dis. Ser. A, 21 (2014), 531-547. Google Scholar [22] Y. Zhang, Y. Zhao, T. Xu and X. Liu, $p$th Moment absolute exponential stability of stochastic control system with Markovian switching, Journal of Industrial and Management Optimization, 12 (2016), 471-486. doi: 10.3934/jimo.2016.12.471. Google Scholar

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##### References:
 [1] W. F. Ames and B. G. Pachpatte, Inequalities for Differential and Integral Equations Academic Press, 197 (1997).Google Scholar [2] G. K. Basak, A. Bisi and M. K. Ghosh, Stability of a random diffusion with linear drift, Journal of Mathematical Analysis and Applications, 202 (1996), 604-622. doi: 10.1006/jmaa.1996.0336. Google Scholar [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. Google Scholar [4] Y. Ji and H. J. Chizeck, Controllability, stabilizability, and continuous-time markovian jump linear quadratic control, Automatic Control IEEE Transactions on, 35 (1990), 777-788. doi: 10.1109/9.57016. Google Scholar [5] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Processes and Their Applications, 79 (1999), 45-67. doi: 10.1016/S0304-4149(98)00070-2. Google Scholar [6] X. Mao, Stochastic functional differential equations with markovian switching, Funct. Differ. Equ, 6 (1999), 375-396. Google Scholar [7] X. Mao, Asymptotic stability for stochastic differential equations with Markovian switching, Funct. Differ. Equ., 9 (2002), 201-220. Google Scholar [8] X. Mao, Exponential stability of stochastic delay interval systems with markovian switching, Automatic Control IEEE Transactions on, 47 (2002), 1604-1612. doi: 10.1109/TAC.2002.803529. Google Scholar [9] 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. Google Scholar [10] X. Mao, J. Lam and L. Huang, Stabilisation of hybrid stochastic differential equations by delay feedback control, Systems Control Letters, 57 (2008), 927-935. doi: 10.1016/j.sysconle.2008.05.002. Google Scholar [11] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching London: Imperial College Press, 2006. doi: 10.1142/p473. Google Scholar [12] X. Mao, A. Matasov and A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli, 6 (2000), 73-90. doi: 10.2307/3318634. Google Scholar [13] X. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica, 49 (2013), 3677-3681. doi: 10.1016/j.automatica.2013.09.005. Google Scholar [14] X. Mao, W. Liu, L. Hu, Q. Luo and J. Lu, Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Systems & Control Letters, 73 (2014), 88-95. doi: 10.1016/j.sysconle.2014.08.011. Google Scholar [15] S. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, 99. Pitman (Advanced Publishing Program), Boston, MA, 1984. Google Scholar [16] Z. Yang, X. Mao and C. Yuan, Comparison theorem of one-dimensional stochastic hybrid delay systems, Systems & Control Letters, 57 (2008), 56-63. doi: 10.1016/j.sysconle.2007.06.014. Google Scholar [17] S. You, W. Liu, J. Lu, X. Mao and Q. 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 [18] C. Yuan and X. Mao, Robust stability and controllability of stochastic differential delay equations with markovian switching, Automatica, 40 (2004), 343-354. doi: 10.1016/j.automatica.2003.10.012. Google Scholar [19] H. Xu, K. L. Teo and X. Liu, Robust stability analysis of guaranteed cost control for impulsive switched systems, IEEE Trans. on Sys. Man. and Cyber. B, 38 (2008), 1419-1422. Google Scholar [20] H. Xu, Y. Chen and K. L. Teo, Global exponential stability of impulsive discrete-time neural networks with time-varying delays, Applied Mathematics and Computation, 217 (2010), 537-544. doi: 10.1016/j.amc.2010.05.087. Google Scholar [21] Y. Zhang, Y. Zhao, M. Shi, H. Shi and C. Liu, The absolute stability of stochastic control system with Markovian switching, Dynam. Cont. Dis. Ser. A, 21 (2014), 531-547. Google Scholar [22] Y. Zhang, Y. Zhao, T. Xu and X. Liu, $p$th Moment absolute exponential stability of stochastic control system with Markovian switching, Journal of Industrial and Management Optimization, 12 (2016), 471-486. doi: 10.3934/jimo.2016.12.471. Google Scholar
Computer simulation of the paths of $r(t)$, $x_1(t)$ and $x_2(t)$ for the discrete-time controlled system (17) using the Euler-Maruyama method with step size $10^{-5}$, delay $\tau = 0.01$ and initial values $r(0) = 1$, $\xi_1 \equiv -20$ and $\xi_2 \equiv 10$ on $[-\tau,0]$
Computer simulation of the paths of $r(t)$, $x_1(t)$ and $x_2(t)$ for the discretetime controlled system (18) with $\alpha = 10^{-4}$ using the Euler-Maruyama method with step size $10^{-5}$, delay $\tau = 0.01$ and initial values $r(0) = 1$, $\xi_1 \equiv -20$ and $\xi_2 \equiv 10$ on $[-\tau,0]$
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