Note on the persistent property of a feedback control system with delays. (English) Zbl 1187.34106
Summary: By developing some new analysis techniques, we show that the following feedback control system of differential equations with delays is permanent.
\[ \begin{aligned} & \frac{dN(t)}{dt}=r(t)N(t)\left[1-\frac{N^2(t-\tau_1(t))}{k^2(t)}-c(t)u(t-\tau_2(t))\right],\\ & \frac{du(t)}{dt}=-a(t)u(t)+b(t)N(t-\tau_1(t)),\end{aligned} \]
where \(\tau_1,\tau_2,a,b,c,r,k\in C(\mathbb{R},(0,+\infty))\) are \(\omega\)-periodic functions, which means that feedback control variable has no influence on the persistent property of the above system.
\[ \begin{aligned} & \frac{dN(t)}{dt}=r(t)N(t)\left[1-\frac{N^2(t-\tau_1(t))}{k^2(t)}-c(t)u(t-\tau_2(t))\right],\\ & \frac{du(t)}{dt}=-a(t)u(t)+b(t)N(t-\tau_1(t)),\end{aligned} \]
where \(\tau_1,\tau_2,a,b,c,r,k\in C(\mathbb{R},(0,+\infty))\) are \(\omega\)-periodic functions, which means that feedback control variable has no influence on the persistent property of the above system.
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