SCIENCE CHINA Information Sciences, Volume 61, Issue 7:
070213(2018)
https://doi.org/10.1007/s11432-017-9297-1

More info

- ReceivedAug 22, 2017
- AcceptedOct 27, 2017
- PublishedJun 13, 2018

Although the mean square stabilization of hybrid systems byfeedback control based on discrete-time observations of state and mode has been studied by severalauthors since 2013,the corresponding almost sure stabilization problemhas received little attention. Recently, Mao was the first to study the almost sure stabilization of a given unstable system$\dot~x(t)=f(x(t))$ by a linear discrete-time stochastic feedback control $Ax([t/\T]\T){\rm~d}B(t)$ (namely the stochastically controlled systemhas the form ${\rm~d}x(t)=f(x(t)){\rm~d}t~+~Ax([t/\T]\T){\rm~d}B(t)$), where $B(t)$ is a scalar Brownian, $\T~>0$, and $[t/\T]$ is the integer part of $t/\T$. In this paper, we consider a much more general problem. That is, we study the almost sure stabilization of a given unstable hybrid system$\dot~x(t)=f(x(t),r(t))$ by nonlinear discrete-time stochastic feedback control $u(x([t/\T]\T),r([t/\T]\T)){\rm~d}B(t)$ (so the stochastically controlled system is a hybrid stochastic system of the form ${\rm~d}x(t)=f(x(t),r(t)){\rm~d}t~+~u(x([t/\T]\T),r([t/\T]\T)){\rm~d}B(t)$), where $B(t)$ is a multi-dimensional Brownian motion and $r(t)$ is a Markov chain.

The authors would like to thank Leverhulme Trust (Grant No. RF-2015-385), Royal Society (Grant No. WM160014, Royal Society Wolfson Research Merit Award), Royal Society and Newton Fund (Grant No. NA160317, Royal Society–Newton Advanced Fellowship), Engineering and Physics Sciences Research Council (Grant No. EP/K503174/1), National Natural Science Foundation of China (Grant Nos. 61503190, 61473334, 61403207), Natural Science Foundation of Jiangsu Province (Grant Nos. BK20150927, BK20131000), and Ministry of Education (MOE) of China (Grant No. MS2014DHDX020) for their financial support. The first author would like to thank Chinese Scholarship Council for awarding him the scholarship to visit the University of Strathclyde for one year.

**Appendix**

Once Assumption

Here, we cite a useful lemma on $M$-matrices.

(1) $A$ is a nonsingular $M$-matrix.

(2) $A$ is semi-positive; that is, there exists $x~>~0$ in $\RR^N$ such that $Ax~>~0$.

(3) $A^{-1}$ exists and its elements are all nonnegative.

(4) All the leading principal minors of $A$ are positive; that is

We also need another classical result.

Now, let us propose a condition that

It was shown in

proof Without loss of generality, we may assume that the state $u=N$ in condition (

By

*[1] *
*
Black
F,
Scholes
M.
Google Scholar
*

*[2] *
*
Berman A, Plemmons R J. Nonnegative Matrices in the Mathematical Sciences. Philadelphia: Society for Industrial and Applied Mathematics, 1994.
Google Scholar
*

*[3] *
*
Krishnamurthy
V,
Wang
X,
Yin
G.
Spreading code optimization and adaptation in CDMA via discrete stochastic approximation.
IEEE Trans Inform Theor,
2004, 50: 1927-1949
CrossRef
Google Scholar
*

*[4] *
*
Luo
Q,
Gong
Y Y,
Jia
C X.
Stability of gene regulatory networks with Lévy noise.
Sci China Inf Sci,
2017, 60: 072204
CrossRef
Google Scholar
*

*[5] *
*
Luo
Q,
Shi
L L,
Zhang
Y T.
Stochastic stabilization of genetic regulatory networks.
Neurocomputing,
2017, 266: 123-127
CrossRef
Google Scholar
*

*[6] *
*
Mao
W H,
Deng
F Q,
Wan
A H.
Robust $H_2/H_{\infty}$ global linearization filter design for nonlinear stochastic time-varying delay systems.
Sci China Inf Sci,
2016, 59: 032204
CrossRef
Google Scholar
*

*[7] *
*
Mao X R. Stability of Stochastic Differential Equations With Respect to Semimartingales. London: Longman Scientific and Technical, 1991.
Google Scholar
*

*[8] *
*
Mao X R. Exponential Stability of Stochastic Differential Equations. New York: Marcel Dekker, 1994.
Google Scholar
*

*[9] *
*
Tang
M,
Meng
Q.
Stochastic evolution equations of jump type with random coefficients: existence, uniqueness and optimal control.
Sci China Inf Sci,
2017, 60: 118202
CrossRef
Google Scholar
*

*[10] *
*
Willsky A S, Levy B C. Stochastic stability research for complex power systems. DOE Contract LIDS MIT Report No. ET-76-C-01-2295, 1979.
Google Scholar
*

*[11] *
*
Sworder
D D,
Rogers
R O.
An LQ-solution to a control problem associated with a solar thermal central receiver.
IEEE Trans Automat Contr,
1983, 28: 971-978
CrossRef
Google Scholar
*

*[12] *
*
Yin
G,
Liu
R H,
Zhang
Q.
Recursive Algorithms for Stock Liquidation: A Stochastic Optimization Approach.
SIAM J Optim,
2002, 13: 240-263
CrossRef
Google Scholar
*

*[13] *
*
Zhang
Q.
Stock trading: an optimal selling rule.
SIAM J Control Optim,
2001, 40: 64-87
CrossRef
Google Scholar
*

*[14] *
*
Anderson W J. Continuous-Time Markov Chains. New York: Springer, 1991.
Google Scholar
*

*[15] *
*
Ji Y, Chizeck H J, Feng X, et al. Stability and control of discrete-time jump linear systems. Contr Theor Adv Tech, 1991, 7: 247--270.
Google Scholar
*

*[16] *
*
Mariton M. Jump Linear Systems in Automatic Control. New York: Marcel Dekker, 1990.
Google Scholar
*

*[17] *
*
Skorohod A V. Asymptotic Methods in the Theory of Stochastic Differential Equations. Providence: American Mathematical Society, 1989.
Google Scholar
*

*[18] *
*
Yin G, Zhang Q. Continuous-Time Markov Chains and Applications: a Singular Perturbation Approach. New York: Springer-Verlag, 1998.
Google Scholar
*

*[19] *
*
Shaikhet L. Stability of stochastic hereditary systems with Markov switching. Theory Stoch Proc, 1996, 2: 180--183.
Google Scholar
*

*[20] *
*
Mao
X R.
Stability of stochastic differential equations with Markovian switching.
Stochastic Processes Appl,
1999, 79: 45-67
CrossRef
Google Scholar
*

*[21] *
*
Arnold
L,
Crauel
H,
Wihstutz
V.
Stabilization of linear systems by noise.
SIAM J Control Optim,
1983, 21: 451-461
CrossRef
Google Scholar
*

*[22] *
*
Basak
G K,
Bisi
A,
Ghosh
M K.
Stability of a random diffusion with linear drift.
J Math Anal Appl,
1996, 202: 604-622
CrossRef
Google Scholar
*

*[23] *
*
Khasminskii R Z. Stochastic Stability of Differential Equations. Leiden: Sijthoff and Noordhoff, 1981.
Google Scholar
*

*[24] *
*
Mao
X R.
Stochastic stabilization and destabilization.
Syst Control Lett,
1994, 23: 279-290
CrossRef
Google Scholar
*

*[25] *
*
Mao X R. Stochastic Differential Equations and Their Applications. 2nd ed. Chichester: Horwood Publishing Limited, 2007.
Google Scholar
*

*[26] *
*
Appleby
J A D,
Mao
X R.
Stochastic stabilisation of functional differential equations.
Syst Control Lett,
2005, 54: 1069-1081
CrossRef
Google Scholar
*

*[27] *
*
Mao X R, Yuan C G. Stochastic Differential Equations With Markovian Switching. London: Imperial College Press, 2006.
Google Scholar
*

*[28] *
*
Mao
X R,
Yin
G G,
Yuan
C.
Stabilization and destabilization of hybrid systems of stochastic differential equations.
Automatica,
2007, 43: 264-273
CrossRef
Google Scholar
*

*[29] *
*
Scheutzow
M K R.
Stabilization and destabilization by noise in the plane.
Stochastic Anal Appl,
1993, 11: 97-113
CrossRef
Google Scholar
*

*[30] *
*
Mao
X R.
Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control.
Automatica,
2013, 49: 3677-3681
CrossRef
Google Scholar
*

*[31] *
*
Mao
X R,
Liu
W,
Hu
L J.
Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations.
Syst Control Lett,
2014, 73: 88-95
CrossRef
Google Scholar
*

*[32] *
*
You
S,
Liu
W,
Lu
J.
Stabilization of hybrid systems by feedback control based on discrete-time state observations.
SIAM J Control Optim,
2015, 53: 905-925
CrossRef
Google Scholar
*

*[33] *
*
Song
G,
Zheng
B C,
Luo
Q.
Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time observations of state and mode.
IET Control Theor Appl,
2017, 11: 301-307
CrossRef
Google Scholar
*

*[34] *
*
Mao
X R.
Almost sure exponential stabilization by discrete-time stochastic feedback control.
IEEE Trans Automat Contr,
2016, 61: 1619-1624
CrossRef
Google Scholar
*

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1