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

Consider the master system

x ˙ 1 =(25α+10)(x 2 -x 1 ),x ˙ 2 =(28-35α)x 1 -x 1 x 3 +(29α-1)x 2 ,x ˙ 3 =x 1 x 2 -(8+α) 3x 3 ·(1)

For α[0,1] system (1) is chaotic, for certain α-values it is related to the Lorenz, Lü and Chen system. Representing (1) in the form x ˙=f(x,α), the authors consider together with (1) the slave system y ˙=f(y,α)+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

e ˙=f ˜(e,y,α)+uwithe=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).

34C28Complex behavior, chaotic systems (ODE)
34H05ODE in connection with control problems