Chaos synchronization of an energy resource system based on linear control (Q708485)

The author considers a system of ordinary differential equations of third order \[ x'(t)=f(x(t)),\quad x\in\mathbb{R}^{3} \] proposed by \textit{M. Sun, L. Tian} and \textit{Y. Fu} [Chaos Solitons Fractals 32, No.~1, 168--180 (2007; Zbl 1133.91524)]. For this energy resource demand-supply system, the following problem is studied: find a function (controller) \(u(x)\) such that the following ``controlled'' system \[ y'=f(y)+u(x-y),\quad x'=f(x),\quad x,y\in\mathbb{R}^{3} \] is synchronized, i.e., \(\|x(t)-y(t)\|\to 0\) as \(t\to \infty\) for all initial conditions. For the specific system, the authors prove that the functions \(u_{1}(x)=\mathrm{diag}\{g,g,g\}x\), \(u_{1}(x)=\mathrm{diag}\{g,0,g\}x\), and \(u_{1}(x)=\mathrm{diag}\{g,0,0\}x\) can be used as controllers if \(g\) is sufficiently large.