# 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 and 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. [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. [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. [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. [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. [6] H. J. Kushner, Stochastic Stability and Control, volume 33 of Mathematics in Science and Engineering, Academic Press, New York, 1967. [7] X. Liao, L. Q. Wang and P. Yu, Stability of Dynamical Systems, Elsevier, 2007. doi: 10.1016/S1574-6917(07)05001-5. [8] X. Liao and P. Yu, Absolute Stability of Nonlinear Control Systems, 2nd edition, Springer, 2008. doi: 10.1007/978-1-4020-8482-9. [9] D. Liberzon, Switching in Systems and Control, Springer, 2003. doi: 10.1007/978-1-4612-0017-8. [10] M. R. Liberzon, Essays on the absolute stability theory, Automation and Remote Control, 67 (2006), 1610-1644. doi: 10.1134/S0005117906100043. [11] A. I. Lurie and V. N. Postnikov, On the theory of stability of control systems, Applied Mathematics and Mechanics, 8 (1944), 246-248. [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. [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. [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. [15] X. Mao, Asymptotic stability for stochastic differential equations with Markovian switching, WSEAS Trans. Circuits, 1 (2002), 68-73. [16] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473. [17] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. doi: 10.1533/9780857099402. [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. [19] V. M. Popov, Absolute stability of nonlinear systems of automatic control, Automation and Remote Control, 22 (1962), 857-875. [20] Z. Sun and S. Ge, Stability Theory of Switched Dynamical Systems, Springer, 2011. doi: 10.1007/978-0-85729-256-8. [21] A. J. Van Der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems, Springer, 2000. doi: 10.1007/BFb0109998. [22] H. Xie, Theory and Application of Absolute Stability, Science Press, Beijing, 1986. [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. [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. [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. [26] V. A. Yakubovich, The solution of certain matrix inequalities in automatic control theory, Soviet Math. Dokl, 3 (1962), 620-623. [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. [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.

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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), 604-622. doi: 10.1006/jmaa.1996.0336. [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. [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. [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. [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. [6] H. J. Kushner, Stochastic Stability and Control, volume 33 of Mathematics in Science and Engineering, Academic Press, New York, 1967. [7] X. Liao, L. Q. Wang and P. Yu, Stability of Dynamical Systems, Elsevier, 2007. doi: 10.1016/S1574-6917(07)05001-5. [8] X. Liao and P. Yu, Absolute Stability of Nonlinear Control Systems, 2nd edition, Springer, 2008. doi: 10.1007/978-1-4020-8482-9. [9] D. Liberzon, Switching in Systems and Control, Springer, 2003. doi: 10.1007/978-1-4612-0017-8. [10] M. R. Liberzon, Essays on the absolute stability theory, Automation and Remote Control, 67 (2006), 1610-1644. doi: 10.1134/S0005117906100043. [11] A. I. Lurie and V. N. Postnikov, On the theory of stability of control systems, Applied Mathematics and Mechanics, 8 (1944), 246-248. [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. [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. [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. [15] X. Mao, Asymptotic stability for stochastic differential equations with Markovian switching, WSEAS Trans. Circuits, 1 (2002), 68-73. [16] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473. [17] X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007. doi: 10.1533/9780857099402. [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. [19] V. M. Popov, Absolute stability of nonlinear systems of automatic control, Automation and Remote Control, 22 (1962), 857-875. [20] Z. Sun and S. Ge, Stability Theory of Switched Dynamical Systems, Springer, 2011. doi: 10.1007/978-0-85729-256-8. [21] A. J. Van Der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems, Springer, 2000. doi: 10.1007/BFb0109998. [22] H. Xie, Theory and Application of Absolute Stability, Science Press, Beijing, 1986. [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. [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. [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. [26] V. A. Yakubovich, The solution of certain matrix inequalities in automatic control theory, Soviet Math. Dokl, 3 (1962), 620-623. [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. [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.
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