×

Impulsive control for differential systems with delay. (English) Zbl 1276.34060

Nonlinear finite-dimensional, nonstationary impulsive systems with constant delay in the state variables are considered. Using Lyapunov functionals sufficient conditions for asymptotic stability are formulated and proved. The special case of linear systems is discussed, and sufficient conditions for asymptotic stability are derived. Similar stability results can be found in the papers [X. Liu, Math. Comput. Modelling 39, No. 4–5, 511–519 (2004; Zbl 1081.93021) and Z. Yang and D. Xu, Comput. Math. Appl. 53, No. 5, 760–769 (2007; Zbl 1133.93362)].

MSC:

34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
34K35 Control problems for functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] YangT. Impulsive control. IEEE Transactions on Automatic Control1999; 44:1081-1083. · Zbl 0954.49022
[2] LiZG, WenCY, SohYC. Analysis and design of impulsive control systems. IEEE Transactions on Automatic Control2001; 46:894-897. · Zbl 1001.93068
[3] BainovDD, SimeonovPS. Systems with Impulse Effect: Stability, Theory and Applications. Halsted: New York, 1989. · Zbl 0683.34032
[4] LakshmikanthamV, BainovDD, SimeonovPS. Theory of Impulsive Differential Equations. World Scientific, Singapore: London, UK, 1989. · Zbl 0719.34002
[5] SunJT, ZhangYP, WuQD. Less conservative conditions for asymptotic stability of impulsive control systems. IEEE Transactions on Automatic Control2003; 48:829-831. · Zbl 1364.93691
[6] YangT, ChuaLO. Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication. IEEE Transactions Circuits and Systems I1997; 44:976-988.
[7] LiZG, WenCY, SohYC, XieWX. The stabilization and synchronization of Chua’s oscillators via impulsive control. IEEE Transactions Circuits and Systems I2001; 48:1351-1355. · Zbl 1024.93052
[8] YuYB, LiK, ZhongQS, YuJB, LiaoXF. Impulsive control of a class of uncertain systems with small time delay. Communications, Circuits and Systems: ICCCAS 20082008; 25:1159-1162.
[9] YuYB, ZhangFL, ZhongQS, LiaoXF, YuJB. Impulsive control of Lurie systems. Computers Mathematics with Applications2008; 56:2806-2813. · Zbl 1165.34380
[10] ZhongQS, BaoJF, YuYB, LiaoXF. Impulsive control for T‐S fuzzy model‐based chaotic systems. Mathematics and Computers in Simulation2008; 79:409-415. · Zbl 1151.93023
[11] LiXD. Exponential stability of Cohen-Grossberg‐type BAM neural networks with time‐varying delays via impulsive control. Neurocomputing2009; 73:525-530.
[12] LiXD, BohnerM. Exponential synchronization of chaotic neural networks with mixed delays and impulsive effects via output coupling with delay feedback. Mathematical and Computer Modelling2010; 52:643-653. · Zbl 1202.34128
[13] LiXD. Exponential stability of Hopfield neural networks with time‐varying delays via impulsive control. Mathematical Methods in the Applied Sciences2010; 33:1596-1604. · Zbl 1247.34026
[14] MahmoudMS, IsmailA. New results on delay‐dependent control of time‐delay systems. IEEE Transactions on Automatic Control2005; 50:95-100. · Zbl 1365.93143
[15] YangZC, XuDY. Stability analysis and design of impulsive control systems with time delay. IEEE Transactions on Automatic Control2007; 52:1448-1454. · Zbl 1366.93276
[16] HuHY, WangZH. Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer‐Verlag: New York, 2002. · Zbl 1035.93002
[17] YangZC, XuDY. Robust stability of uncertain impulsive control systems with time‐varying delay. Computers Mathematics with Applications2007; 53:760-769. · Zbl 1133.93362
[18] LiuXZ. Stability of impulsive control systems with time delay. Mathematical and Computer Modelling2004; 39:511-519. · Zbl 1081.93021
[19] WangQ, LiuXZ. Impulsive stabilization of delay differential systems via the Lyapunov‐Razumikhin method. Applied Mathematics Letters1995; 193:923-941. · Zbl 0837.34076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.