Synchronous control of chaotic complex systems (Q2706705)

Consider the chaotic nonlinear system NEWLINE\[NEWLINE{dx\over dt}= f(x),\quad x\in \mathbb{R}^n,\quad t\in [0,\infty),\tag{1}NEWLINE\]NEWLINE with a target trajectory \(y^*(t)\) satisfying the motion equation NEWLINE\[NEWLINE{dy\over dt}= g(y),\quad y\in \mathbb{R}^n,NEWLINE\]NEWLINE where \(f,g: \mathbb{R}^n\to \mathbb{R}^n\) is continuous, \(f(0)= g(0)= 0\). In \textit{L. Kocarev}, \textit{A. Shane} and \textit{L. O. Ehna} [Int. J. Bifurcation Chaos, Appl. Sci. Eng. 3, 479-483 (1993; Zbl 0873.34046)], using the linear feedback controller \(u= K[y^*(t)- x]\), where \(K\) is a diagonal constant matrix with positive entries, it follows that the controlled system has the form NEWLINE\[NEWLINE{dx\over dt}= f(x)+ K[y^*(t)- x],\quad x\in \mathbb{R}^n,\quad t\in [0,\infty).\tag{\(*\)}NEWLINE\]NEWLINE Two theorems are given therein for the case when \(f\equiv g\) and \(f\not\equiv g\), to ensure that the relation NEWLINE\[NEWLINE\lim_{t\to\infty} [x(t)- y^*(t)]= 0NEWLINE\]NEWLINE holds for all solution \(x(t)\) of (1). But, however, as indicated by a contradictory example in the paper, the proofs given in the above-mentioned article are not rigorous enough.NEWLINENEWLINENEWLINEApplying Gronwall's integral inequality and an inverse theorem in the theory of Lyapunov direct methods, the author establishes two sufficient conditions for the desired synchronous control when \(f\equiv g\). In addition, it is shown that when \(f\not\equiv g\) the synchronous control can not be implemented in the same way. An application of the results obtained to the synchronization of Lorenz system is also presented.