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Finite-time synchronization of uncertain unified chaotic systems based on CLF. (English) Zbl 1183.34072
Consider the master system
\[ \begin{aligned} \dot x_1 & = (25\alpha+ 10)(x_2- x_1),\\ \dot x_2 & = (28-35\alpha) x_1- x_1 x_3+ (29\alpha- 1)x_2,\\ \dot x_3 & = x_1 x_2- {(8+ \alpha)\over 3} x_3.\end{aligned}\tag{1} \] 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).

MSC:
34D06 Synchronization of solutions to ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34H05 Control problems involving ordinary differential equations
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