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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.

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