<p style='text-indent:20px;'>This paper deals with the analysis of the internal controllability with constraint of positive kind of a quasilinear parabolic PDE. We prove two results about this PDE: First, we prove a global steady state constrained controllability result. For this purpose, we employ the called "stair-case method". And second, we prove a global trajectory constrained controllability result. For this purpose, we employ the well-known "stabilization property" in <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> norms. Furthermore, for both results an important argument is needed: the exact local controllability to trajectories. Then we prove the positivity of the minimal controllability time using arguments of comparison principle. Some additional comments and open problems concerning other systems are presented.</p>

中文翻译：

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拟线性抛物线偏微分方程正约束下的可控性

<p style='text-indent:20px;'>本文研究了拟线性抛物线偏微分方程带正类约束的内部可控性分析。我们证明了这个 PDE 的两个结果：首先，我们证明了全局稳态约束可控性结果。为此，我们采用了所谓的“阶梯法”。其次，我们证明了全局轨迹约束可控性结果。为此，我们在 <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math 中使用了众所周知的“稳定属性” ></inline-formula> 规范。此外，对于这两个结果，都需要一个重要的论据：对轨迹的精确局部可控性。然后我们利用比较原理论证证明了最小可控时间的积极性。提出了一些关于其他系统的附加评论和未解决的问题。</p>