<p style='text-indent:20px;'>This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-dimensional linear reaction-convection-diffusion equation with <inline-formula><tex-math id="M2">\begin{document}$ N\geq 1 $\end{document}</tex-math></inline-formula> and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state.</p>

中文翻译：

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具有输入延迟的抛物线系统的反馈稳定

<p style='text-indent:20px;'>这项工作致力于稳定具有恒定延迟的有限维控制的抛物线系统。我们的主要结果表明，Fattorini-Hautus 准则产生了这种反馈控制的存在，就像在没有延迟的稳定的情况下一样。证明包括将系统分成有限维不稳定部分和稳定无限维部分，并对有限维系统应用Artstein变换以消除控制中的延迟。使用我们的抽象结果，我们可以证明具有恒定延迟的抛物线系统稳定性的新结果： <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}< /tex-math></inline-formula> 具有 <inline-formula><tex-math id="M2">\begin{document}$ N\geq 1 $\end{document}</tex-math></inline 的一维线性反应-对流-扩散方程-formula> 和 Oseen 系统。最后，我们通过证明 Navier-Stokes 系统在稳态附近的局部反馈分布稳定，证明了该理论可用于稳定具有输入延迟的非线性抛物线系统。</p>