American Institute of Mathematical Sciences

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

 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.
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.  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.   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.  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).  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.  doi: 10.1007/BF01057179.  Google Scholar [6] H. J. Kushner, Stochastic Stability and Control, volume 33 of Mathematics in Science and Engineering,, Academic Press, (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, (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.  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.   Google Scholar [12] A. K. Mahalanabis and S. Purkayastha, Frequency-domain criteria for stability of a class of nonlinear stochastic systems,, Automatic Control, 18 (1973), 266.   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.  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.  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.   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.  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.   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, (1986).   Google Scholar [23] H. Xu and K. L. Teo, Exponential stability with-gain condition of nonlinear impulsive switched systems,, Automatic Control, 55 (2010), 2429.  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.  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).  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.   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.  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).   Google Scholar

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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.  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.   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.  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).  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.  doi: 10.1007/BF01057179.  Google Scholar [6] H. J. Kushner, Stochastic Stability and Control, volume 33 of Mathematics in Science and Engineering,, Academic Press, (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, (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.  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.   Google Scholar [12] A. K. Mahalanabis and S. Purkayastha, Frequency-domain criteria for stability of a class of nonlinear stochastic systems,, Automatic Control, 18 (1973), 266.   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.  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.  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.   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.  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.   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, (1986).   Google Scholar [23] H. Xu and K. L. Teo, Exponential stability with-gain condition of nonlinear impulsive switched systems,, Automatic Control, 55 (2010), 2429.  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.  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).  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.   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.  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).   Google Scholar
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