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Finite-time synchronization of uncertain unified chaotic systems based on CLF. (English) Zbl 1183.34072

Consider the master system

$\begin{array}{cc}\hfill {\stackrel{˙}{x}}_{1}& =\left(25\alpha +10\right)\left({x}_{2}-{x}_{1}\right),\hfill \\ \hfill {\stackrel{˙}{x}}_{2}& =\left(28-35\alpha \right){x}_{1}-{x}_{1}{x}_{3}+\left(29\alpha -1\right){x}_{2},\hfill \\ \hfill {\stackrel{˙}{x}}_{3}& ={x}_{1}{x}_{2}-\frac{\left(8+\alpha \right)}{3}{x}_{3}·\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

For $\alpha \in \left[0,1\right]$ system (1) is chaotic, for certain $\alpha$-values it is related to the Lorenz, Lü and Chen system. Representing (1) in the form $\stackrel{˙}{x}=f\left(x,\alpha \right)$, the authors consider together with (1) the slave system $\stackrel{˙}{y}=f\left(y,\alpha \right)+u$. 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

$\stackrel{˙}{e}=\stackrel{˜}{f}\left(e,y,\alpha \right)+u\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}e=y-x$

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 34C28 Complex behavior, chaotic systems (ODE) 34H05 ODE in connection with control problems