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

Yuyun Zhao; Yi Zhang; Tao Xu; Ling Bai; Qian Zhang

doi:10.3934/dcdsb.2017011

Article Contents

2017, Volume 22, Issue 1: 209-226. Doi: 10.3934/dcdsb.2017011

This issue Previous Article Improved results on exponential stability of discrete-time switched delay systems Next Article

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
* Corresponding author: Yi Zhang

Received: July 31, 2015

Revised: April 30, 2016

Published: November 2017

Abstract Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

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.

Keywords:

$p$th moment exponential stability,
stochastic functional differential equation,
Markovian switching,
discrete-time state observations.

Mathematics Subject Classification: 34H15, 34K50, 93E03, 93E15.

Citation:

Full Text(HTML)

Figure 1. 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 $[-\tau,0]$

Download: Full-size image PowerPoint slide

Figure 2. 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 $[-\tau,0]$

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

	W. F. Ames and B. G. Pachpatte, Inequalities for Differential and Integral Equations Academic Press, 197 (1997).
	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.
	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.
	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.
	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.
	X. Mao , Stochastic functional differential equations with markovian switching, Funct. Differ. Equ, 6 (1999) , 375-396.
	X. Mao , Asymptotic stability for stochastic differential equations with Markovian switching, Funct. Differ. Equ., 9 (2002) , 201-220.
	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.
	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.
	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.
	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching London: Imperial College Press, 2006. doi: 10.1142/p473.
	X. Mao , A. Matasov and A. B. Piunovskiy , Stochastic differential delay equations with Markovian switching, Bernoulli, 6 (2000) , 73-90. doi: 10.2307/3318634.
	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.
	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.
	S. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, 99. Pitman (Advanced Publishing Program), Boston, MA, 1984.
	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.
	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.
	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.
	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.
	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.
	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.
	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.