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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).

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
Full Text: DOI
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