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$p$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 |
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. |
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. |
[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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