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Chaos synchronization of an energy resource system based on linear control. (English) Zbl 1216.34061
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 M. Sun, L. Tian and 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.

MSC:
34H10 Chaos control for problems involving ordinary differential equations
34H05 Control problems involving ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
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