Almost sure exponential stability and stochastic stabilization of impulsive stochastic differential delay equations. (English) Zbl 07878497
Summary: In this paper, we mainly study the almost sure exponential stability of impulsive stochastic differential delay equations (ISDDEs) with bounded variable delays. The main technique is to compare ISDDEs with corresponding impulsive stochastic differential equations (ISDEs) without delay, to obtain the upper bound \(\tau^\ast\) of delays that ISDDEs can maintain stability by accurate calculation. The results show that if the corresponding ISDEs are almost surely exponentially stable, then the ISDDEs are also almost surely exponentially stable as long as these delays are less than \(\tau^\ast \). In addition, the stability theory established in this paper can be applied to noise stabilization based on sampled-data observations of a class of unstable impulsive systems. Taking impulsive Lurie systems (ILSs) as an example, we discuss the design of noise stabilization strategies.
MSC:
| 93D23 | Exponential stability |
| 93E15 | Stochastic stability in control theory |
| 93C27 | Impulsive control/observation systems |
| 93C57 | Sampled-data control/observation systems |
| 93C23 | Control/observation systems governed by functional-differential equations |
| 34K45 | Functional-differential equations with impulses |
| 34K50 | Stochastic functional-differential equations |
Keywords:
almost sure exponential stability; impulsive stochastic differential delay equations; noise stabilization; sampled-data observationsReferences:
| [1] | Mil’man, U.; Myshkis, On the stability of motion in the presence of impulses, Sibirsk. Mat., 1, 233-237, 1960 · Zbl 1358.34022 |
| [2] | Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations, 1989, World Scientific: World Scientific Singapore · Zbl 0719.34002 |
| [3] | Bainov, D. D.; Simeonov, P. S., Systems with Impulse Effect: Stability, Theory and Applications, 1989, Ellis Horwood: Ellis Horwood Chichester · Zbl 0683.34032 |
| [4] | Benchohra, M.; Henderson, J.; Ntouyas, S., Impulsive Differential Equations and Inclusions, 2006, Hindawi Publishing Corporation · Zbl 1130.34003 |
| [5] | Yang, T.; Chua, L. O., Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE Trans. Circuits Syst., 44, 10, 976-988, 1997 |
| [6] | Sun, J. T.; Zhang, Y. P.; Wu, Q. D., Impulsive control for the stabilization and synchronization of Lorenz systems, Phys. Lett. A, 298, 2-3, 153-160, 2002 · Zbl 0995.37021 |
| [7] | Khadra, A.; Liu, X. Z.; Shen, X. M., Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses, IEEE Trans. Automat. Control, 54, 4, 923-928, 2009 · Zbl 1367.34084 |
| [8] | Lu, J. Q.; Ho, D. W.C.; Cao, J. D., A unified synchronization criterion impulsive dynamical networks, Automatica, 46, 7, 1215-1221, 2010 · Zbl 1194.93090 |
| [9] | Guan, K. Z.; Luo, R., Finite-time stability of impulsive pantograph systems with applications, Systems Control Lett., 157, Article 105054 pp., 2021 · Zbl 1480.93365 |
| [10] | Hasminskii, R. Z., Stochastic Stability of Differential Equations, 1981, Noordhoff, Alphen aan den Rijn: Noordhoff, Alphen aan den Rijn Netherlands |
| [11] | Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions, 1992, Cambridge University Press · Zbl 0761.60052 |
| [12] | Protter, P. E., (Stochastic Integration and Differential Equations, 2004, Springer) · Zbl 1041.60005 |
| [13] | Mao, X. R., Stochastic stabilisation and destabilisation, Systems Control Lett., 23, 279-290, 1994 · Zbl 0820.93071 |
| [14] | Mao, X. R., (Stochastic Differnetial Equation and their Applications, 2007, Horwood Pub.: Horwood Pub. New York) · Zbl 1138.60005 |
| [15] | Liu, B., Stability of solutions for stochastic impulsive systems via comparison approach, IEEE Trans. Automat. Control, 53, 9, 2128-2133, 2008 · Zbl 1367.93523 |
| [16] | Alwan, M. S.; Liu, X. Z., Theory of Hybrid Systems: Deterministic and Stochastic, 2018, Springer Nature Singapore Pte Ltd. and Higher Education Press: Springer Nature Singapore Pte Ltd. and Higher Education Press Beijing · Zbl 1439.34001 |
| [17] | Yao, M.; Wei, G. L.; Ding, D. R.; Li, W. Y., Output-feedback control for stochastic impulsive systems under Round-Robin protocol, Automatica, 143, Article 110394 pp., 2022 · Zbl 1497.93217 |
| [18] | Tran, K. Q.; Nguyen, D. H., Exponential stability of impulsive stochastic differential equations with Markovian switching, Systems Control Lett., 162, Article 105178 pp., 2022 · Zbl 1485.93496 |
| [19] | Liu, G. D.; Wang, X. H.; Meng, X. Z.; Gao, S. J., Extinction and persistence in mean of a novel delay impulsive stochastic infected predator-prey system with jumps, Complexity, 2017 · Zbl 1407.92112 |
| [20] | Yang, J.; Zhong, S. M.; Luo, W. P., Mean square stability analysis of impulsive stochastic differential equations with delays, J. Comput. Appl. Math., 216, 474-483, 2008 · Zbl 1142.93035 |
| [21] | Li, X. D.; Zhu, Q. X.; O’Regan, D., \(p\) th Moment exponential stability of impulsive stochastic functional differential equations and application to control problems of NNs, J. Franklin Inst., 351, 4435-4456, 2014 · Zbl 1395.93566 |
| [22] | Pan, L. J.; Cao, J. D., Exponential stability of impulsive stochastic functional differential equations, J. Math. Anal. Appl., 382, 2, 672-685, 2011 · Zbl 1222.60043 |
| [23] | Tran, K. Q.; Yin, G., Exponential stability of stochastic functional differential equations with impulsive perturbations and Markovian switching, Systems Control Lett., 173, Article 105457 pp., 2023 · Zbl 1530.93422 |
| [24] | Peng, S.; Deng, F., New criteria on \(p\) th moment input-to-state stability of impulsive stochastic delayed differential systems, IEEE Trans. Automat. Control, 62, 7, 3573-3579, 2017 · Zbl 1370.34132 |
| [25] | Wang, Z. Y.; Zhu, Q. X., Two categories of new criteria of \(p\) th moment stability for switching and impulsive stochastic delayed functional differential equation with markovian switching, J. Franklin Inst., 360, 3459-3478, 2022 · Zbl 1508.93230 |
| [26] | Cheng, P.; Deng, F. Q.; Yao, F. Q., Exponential stability analysis of impulsive stochastic functional differential systems with delayed impulses, Commun. Nonlinear Sci. Numer. Simul., 19, 2104-2114, 2014 · Zbl 1457.34120 |
| [27] | Kao, Y. G.; Zhu, Q. X.; Qi, W. H., Exponential stability and instability of impulsive stochastic functional differential equations with Markovian switching, Appl. Math. Comput., 271, 15, 795-804, 2015 · Zbl 1410.34213 |
| [28] | Guo, Y. X.; Zhu, Q. X.; Wang, F., Stability analysis of impulsive stochastic functional differential equations, Commun. Nonlinear Sci. Numer. Simul., 82, Article 105013 pp., 2020 · Zbl 1457.34121 |
| [29] | Kuang, H. Y.; Li, J. L., Stability of stochastic functional differential equations with impulses, Appl. Math. Lett., 145, Article 108735 pp., 2023 · Zbl 1523.34089 |
| [30] | Mao, X. R.; Yuan, C. G., Stochastic Differential Equations with Markovian Switching, 2006, Imperial College Press: Imperial College Press UK · Zbl 1126.60002 |
| [31] | Yin, G.; Zhu, C., Hybrid Switching Diffusions: Properties and Applications, 2010, Springer, New York: Springer, New York USA · Zbl 1279.60007 |
| [32] | Mao, X. R.; Yin, G.; Yuan, C. G., Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43, 264-273, 2007 · Zbl 1111.93082 |
| [33] | Deng, F. Q.; Luo, Q.; Mao, X. R., Stochastic stabilization of hybrid differential equations, Automatica, 48, 2321-2328, 2012 · Zbl 1257.93103 |
| [34] | Cheng, P.; Deng, F.; Yao, F., Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects, Nonlinear Anal. Hybrid Syst., 30, 106-117, 2018 · Zbl 1408.93135 |
| [35] | Mao, X. R., Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Trans. Automat. Control, 61, 6, 1619-1624, 2016 · Zbl 1359.93517 |
| [36] | Song, G. F.; Lu, Z. Y.; Zheng, B. C.; Mao, X. R., Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state, Sci. China Inf. Sci., 61, Article 070213 pp., 2018 |
| [37] | Liu, X.; Cheng, P., Almost sure exponential stability and stabilization of hybrid stochastic functional differential equations with Lévy noise, J. Appl. Math. Comput., 69, 3433-3458, 2023 · Zbl 1536.93727 |
| [38] | Alwan, M. S.; Liu, X. Z.; Xie, W. C., Existence, continuation, and uniqueness problems of stochastic impulsive systems with time delay, J. Franklin Inst., 347, 1317-1333, 2010 · Zbl 1205.60107 |
| [39] | Guo, Q.; Mao, X. R.; Yue, R. X., Almost sure exponential stability of stochastic differential delay equations, SIAM J. Control Optim., 54, 4, 1919-1933, 2016 · Zbl 1343.60072 |
| [40] | Yu, Y. B.; Zhang, F. L.; Zhong, Q. S.; Liao, X. F.; Yu, J. B., Impulsive control of Lurie systems, Comput. Math. Appl., 56, 2806-2813, 2008 · Zbl 1165.34380 |
| [41] | Li, X. D.; Chen, W. H.; Zheng, W. X.; Wang, Q. G., Instability and unboundedness analysis for impulsive differential systems with applications to Lurie control systems, Int. J. Control Autom. Syst., 16, 4, 1521-1531, 2018 |
| [42] | Li, D. Q.; Cheng, P.; He, S. P., Exponential stability of hybrid stochastic functional differential systems with delayed impulsive effects: average impulsive interval approach, Math. Methods Appl. Sci., 40, 4197-4210, 2017 · Zbl 1370.34130 |
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