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Zbl 1183.34072
Wang, Hua; Han, Zheng-Zhi; Xie, Qi-Yue; Zhang, Wei
Finite-time synchronization of uncertain unified chaotic systems based on CLF. (English)
[J] Nonlinear Anal., Real World Appl. 10, No. 5, 2842-2849 (2009). ISSN 1468-1218

Consider the master system $$\aligned \dot x_1 & = (25\alpha+ 10)(x_2- x_1),\\ \dot x_2 & = (28-35\alpha) x_1- x_1 x_3+ (29\alpha- 1)x_2,\\ \dot x_3 & = x_1 x_2- {(8+ \alpha)\over 3} x_3.\endaligned\tag1$$ For $\alpha\in [0,1]$ system (1) is chaotic, for certain $\alpha$-values it is related to the Lorenz, Lü and Chen system. Representing (1) in the form $\dot x= f(x,\alpha)$, the authors consider together with (1) the slave system $\dot y= f(y,\alpha)+ u$. The goal of the authors is to find a control $u$ such that the slave system synchronizes the master system in finite time, that is, the corresponding error system $$\dot e=\widetilde f(e,y,\alpha)+ u\text{ with }e= y- x$$ has the property that their solutions tend to zero in a finite time. Of course, this requires that the error system is not Lipschitzian in $e$. The authors construct such a control by means of a control Lyapunov function. Moreover, they show that this control is robust against perturbations of some coefficients of (1).
[Klaus R. Schneider (Berlin)]

MSC 2000:: *34D06
34C28 Other types of "recurrent" solutions of ODE
34H05 ODE in connection with control problems

Keywords: finite-time synchronization; unified chaotic systems; uncertain parameters; control Lyapunov function

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