American Institute of Mathematical Sciences

Advanced
April  2016, 12(2): 471-486. doi: 10.3934/jimo.2016.12.471

$p$th Moment absolute exponential stability of stochastic control system with Markovian switching

Yi Zhang 1, , Yuyun Zhao 1, , Tao Xu 1, and  Xin Liu 1,

1. 

Department of Mathematics, College of Science, China University of Petroleum, Beijing 102249, China, China, China, China

Received  September 2014 Revised  February 2015 Published  June 2015

In this paper we discuss the $p$th moment absolute exponential stability of stochastic control system with Markovian switching. We first give a new concept of $p$th moment absolute exponential stability, then we establish some theorems under different hypotheses to guarantee the system $p$th moment absolutely exponentially stable. These sufficient conditions in our theorems are algebraic criteria in terms of matrix inequalities, and we introduce an $M$-method with MATLAB to compute them. Finally, some examples are given to illustrate our results.
Keywords: Markovian switching., $p$th moment absolute exponential stability, Absolute stability, stochastic differential equation, stochastic control systems.
Mathematics Subject Classification: Primary: 60H10, 34F05, 93E03, 93E15; Secondary: 60J27, 60G1.
Citation: Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471
References:
[1]

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

[2]

V. A. Brusin and V. A. Ugrinovskii, Stochastic stability of a class of nonlinear differential equations of Ito type, Siberian Mathematical Journal, 28 (1987), 381-393.  Google Scholar

[3]

C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighboring extremal solution for an optimal switched impulsive control problem, Journal of Industrial and Management Optimization, 8 (2012), 591-609. doi: 10.3934/jimo.2012.8.591.  Google Scholar

[4]

R. E. Kalman, Lyapunov functions for the problem of Lur'e in automatic control, Proceedings of the National Academy of Sciences of the United States of America, 49 (1963), 201. doi: 10.1073/pnas.49.2.201.  Google Scholar

[5]

D. G. Korenevskii, Algebraic criteria for absolute (relative to nonlinearity) stability of stochastic automatic control systems with nonlinear feedback, Ukrainian Mathematical Journal, 40 (1988), 616-621. doi: 10.1007/BF01057179.  Google Scholar

[6]

H. J. Kushner, Stochastic Stability and Control, volume 33 of Mathematics in Science and Engineering, Academic Press, New York, 1967.  Google Scholar

[7]

X. Liao, L. Q. Wang and P. Yu, Stability of Dynamical Systems, Elsevier, 2007. doi: 10.1016/S1574-6917(07)05001-5.  Google Scholar

[8]

X. Liao and P. Yu, Absolute Stability of Nonlinear Control Systems, 2nd edition, Springer, 2008. doi: 10.1007/978-1-4020-8482-9.  Google Scholar

[9]

D. Liberzon, Switching in Systems and Control, Springer, 2003. doi: 10.1007/978-1-4612-0017-8.  Google Scholar

[10]

M. R. Liberzon, Essays on the absolute stability theory, Automation and Remote Control, 67 (2006), 1610-1644. doi: 10.1134/S0005117906100043.  Google Scholar

[11]

A. I. Lurie and V. N. Postnikov, On the theory of stability of control systems, Applied Mathematics and Mechanics, 8 (1944), 246-248. Google Scholar

[12]

A. K. Mahalanabis and S. Purkayastha, Frequency-domain criteria for stability of a class of nonlinear stochastic systems, Automatic Control, IEEE Transactions on, 18 (1973), 266-270.  Google Scholar

[13]

L. Li, Y. Gao and H. Wang, Second order sufficient optimality conditions for hybrid control problems with state jump, Journal of Industrial and Management Optimization, 11 (2015), 329-343. doi: 10.3934/jimo.2015.11.329.  Google Scholar

[14]

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

[15]

X. Mao, Asymptotic stability for stochastic differential equations with Markovian switching, WSEAS Trans. Circuits, 1 (2002), 68-73.  Google Scholar

[16]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473.  Google Scholar

[17]

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. doi: 10.1533/9780857099402.  Google Scholar

[18]

P. V. Pakshin and V. A. Ugrinovskii, Stochastic problems of absolute stability, Automation and Remote Control, 67 (2006), 1811-1846. doi: 10.1134/S0005117906110051.  Google Scholar

[19]

V. M. Popov, Absolute stability of nonlinear systems of automatic control, Automation and Remote Control, 22 (1962), 857-875.  Google Scholar

[20]

Z. Sun and S. Ge, Stability Theory of Switched Dynamical Systems, Springer, 2011. doi: 10.1007/978-0-85729-256-8.  Google Scholar

[21]

A. J. Van Der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems, Springer, 2000. doi: 10.1007/BFb0109998.  Google Scholar

[22]

H. Xie, Theory and Application of Absolute Stability, Science Press, Beijing, 1986. Google Scholar

[23]

H. Xu and K. L. Teo, Exponential stability with-gain condition of nonlinear impulsive switched systems, Automatic Control, IEEE Transactions on, 55 (2010), 2429-2433. doi: 10.1109/TAC.2010.2060173.  Google Scholar

[24]

H. Xu, K. L. Teo and W. Gui, Necessary and sufficient conditions for stability of impulsive switched linear systems, Discrete and Continuous Dynamical Systems-Series B, 16 (2011), 1185-1195. doi: 10.3934/dcdsb.2011.16.1185.  Google Scholar

[25]

X. Xie, H. Xu and R. Zhang, Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control, Abstract and Applied Analysis, 2014 (2014), Hindawi Publishing Corporation. doi: 10.1155/2014/126836.  Google Scholar

[26]

V. A. Yakubovich, The solution of certain matrix inequalities in automatic control theory, Soviet Math. Dokl, 3 (1962), 620-623. Google Scholar

[27]

Y. Zhang, M. Wang, H. Xu and K. L. Teo, Global stabilization of switched control systems with time delay, Nonlinear Analysis: Hybrid Systems, 14 (2014), 86-98. doi: 10.1016/j.nahs.2014.05.004.  Google Scholar

[28]

Y. Zhang, Y. Zhao, H. Xu, H. Shi and K. L. Teo, On boundedness and attractiveness of nonlinear switched delay systems, In Abstract and Applied Analysis, 2013 (2013), Hindawi Publishing Corporation.  Google Scholar

show all references

References:
[1]

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

[2]

V. A. Brusin and V. A. Ugrinovskii, Stochastic stability of a class of nonlinear differential equations of Ito type, Siberian Mathematical Journal, 28 (1987), 381-393.  Google Scholar

[3]

C. Jiang, K. L. Teo, R. Loxton and G. R. Duan, A neighboring extremal solution for an optimal switched impulsive control problem, Journal of Industrial and Management Optimization, 8 (2012), 591-609. doi: 10.3934/jimo.2012.8.591.  Google Scholar

[4]

R. E. Kalman, Lyapunov functions for the problem of Lur'e in automatic control, Proceedings of the National Academy of Sciences of the United States of America, 49 (1963), 201. doi: 10.1073/pnas.49.2.201.  Google Scholar

[5]

D. G. Korenevskii, Algebraic criteria for absolute (relative to nonlinearity) stability of stochastic automatic control systems with nonlinear feedback, Ukrainian Mathematical Journal, 40 (1988), 616-621. doi: 10.1007/BF01057179.  Google Scholar

[6]

H. J. Kushner, Stochastic Stability and Control, volume 33 of Mathematics in Science and Engineering, Academic Press, New York, 1967.  Google Scholar

[7]

X. Liao, L. Q. Wang and P. Yu, Stability of Dynamical Systems, Elsevier, 2007. doi: 10.1016/S1574-6917(07)05001-5.  Google Scholar

[8]

X. Liao and P. Yu, Absolute Stability of Nonlinear Control Systems, 2nd edition, Springer, 2008. doi: 10.1007/978-1-4020-8482-9.  Google Scholar

[9]

D. Liberzon, Switching in Systems and Control, Springer, 2003. doi: 10.1007/978-1-4612-0017-8.  Google Scholar

[10]

M. R. Liberzon, Essays on the absolute stability theory, Automation and Remote Control, 67 (2006), 1610-1644. doi: 10.1134/S0005117906100043.  Google Scholar

[11]

A. I. Lurie and V. N. Postnikov, On the theory of stability of control systems, Applied Mathematics and Mechanics, 8 (1944), 246-248. Google Scholar

[12]

A. K. Mahalanabis and S. Purkayastha, Frequency-domain criteria for stability of a class of nonlinear stochastic systems, Automatic Control, IEEE Transactions on, 18 (1973), 266-270.  Google Scholar

[13]

L. Li, Y. Gao and H. Wang, Second order sufficient optimality conditions for hybrid control problems with state jump, Journal of Industrial and Management Optimization, 11 (2015), 329-343. doi: 10.3934/jimo.2015.11.329.  Google Scholar

[14]

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

[15]

X. Mao, Asymptotic stability for stochastic differential equations with Markovian switching, WSEAS Trans. Circuits, 1 (2002), 68-73.  Google Scholar

[16]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473.  Google Scholar

[17]

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. doi: 10.1533/9780857099402.  Google Scholar

[18]

P. V. Pakshin and V. A. Ugrinovskii, Stochastic problems of absolute stability, Automation and Remote Control, 67 (2006), 1811-1846. doi: 10.1134/S0005117906110051.  Google Scholar

[19]

V. M. Popov, Absolute stability of nonlinear systems of automatic control, Automation and Remote Control, 22 (1962), 857-875.  Google Scholar

[20]

Z. Sun and S. Ge, Stability Theory of Switched Dynamical Systems, Springer, 2011. doi: 10.1007/978-0-85729-256-8.  Google Scholar

[21]

A. J. Van Der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems, Springer, 2000. doi: 10.1007/BFb0109998.  Google Scholar

[22]

H. Xie, Theory and Application of Absolute Stability, Science Press, Beijing, 1986. Google Scholar

[23]

H. Xu and K. L. Teo, Exponential stability with-gain condition of nonlinear impulsive switched systems, Automatic Control, IEEE Transactions on, 55 (2010), 2429-2433. doi: 10.1109/TAC.2010.2060173.  Google Scholar

[24]

H. Xu, K. L. Teo and W. Gui, Necessary and sufficient conditions for stability of impulsive switched linear systems, Discrete and Continuous Dynamical Systems-Series B, 16 (2011), 1185-1195. doi: 10.3934/dcdsb.2011.16.1185.  Google Scholar

[25]

X. Xie, H. Xu and R. Zhang, Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control, Abstract and Applied Analysis, 2014 (2014), Hindawi Publishing Corporation. doi: 10.1155/2014/126836.  Google Scholar

[26]

V. A. Yakubovich, The solution of certain matrix inequalities in automatic control theory, Soviet Math. Dokl, 3 (1962), 620-623. Google Scholar

[27]

Y. Zhang, M. Wang, H. Xu and K. L. Teo, Global stabilization of switched control systems with time delay, Nonlinear Analysis: Hybrid Systems, 14 (2014), 86-98. doi: 10.1016/j.nahs.2014.05.004.  Google Scholar

[28]

Y. Zhang, Y. Zhao, H. Xu, H. Shi and K. L. Teo, On boundedness and attractiveness of nonlinear switched delay systems, In Abstract and Applied Analysis, 2013 (2013), Hindawi Publishing Corporation.  Google Scholar

[1]

Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697
[2]

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
[3]

Xiaojin Huang, Hongfu Yang, Jianhua Huang. Consensus stability analysis for stochastic multi-agent systems with multiplicative measurement noises and Markovian switching topologies. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021024
[4]

Litan Yan, Wenyi Pei, Zhenzhong Zhang. Exponential stability of SDEs driven by fBm with Markovian switching. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6467-6483. doi: 10.3934/dcds.2019280
[5]

Yaozhong Hu, David Nualart, Xiaobin Sun, Yingchao Xie. Smoothness of density for stochastic differential equations with Markovian switching. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3615-3631. doi: 10.3934/dcdsb.2018307
[6]

Henri Schurz. Moment attractivity, stability and contractivity exponents of stochastic dynamical systems. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 487-515. doi: 10.3934/dcds.2001.7.487
[7]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[8]

Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157
[9]

Syvert P. Nørsett, Andreas Asheim. Regarding the absolute stability of Størmer-Cowell methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1131-1146. doi: 10.3934/dcds.2014.34.1131
[10]

Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041
[11]

Yong He. Switching controls for linear stochastic differential systems. Mathematical Control & Related Fields, 2020, 10 (2) : 443-454. doi: 10.3934/mcrf.2020005
[12]

Weijun Zhan, Qian Guo, Yuhao Cong. The truncated Milstein method for super-linear stochastic differential equations with Markovian switching. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021201
[13]

Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control & Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019
[14]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157
[15]

Tomás Caraballo, Carlos Ogouyandjou, Fulbert Kuessi Allognissode, Mamadou Abdoul Diop. Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 507-528. doi: 10.3934/dcdsb.2019251
[16]

Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021040
[17]

Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control & Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359
[18]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[19]

Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325
[20]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4887-4905. doi: 10.3934/dcdsb.2020317

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (103)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy