<p style='text-indent:20px;'>This paper considers two types of boundary control problems for linear transport equations. The first one shows that transport solutions on a subdomain of a domain <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula> can be controlled exactly from incoming boundary conditions for <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula> under appropriate convexity assumptions. This is in contrast with the only approximate control one typically obtains for elliptic equations by an application of a unique continuation property, a property which we prove does not hold for transport equations. We also consider the control of an outgoing solution from incoming conditions, a transport notion similar to the Dirichlet-to-Neumann map for elliptic equations. We show that for well-chosen coefficients in the transport equation, this control may not be possible. In such situations and by (Fredholm) duality, we obtain the existence of non-trivial incoming conditions that are compatible with vanishing outgoing conditions.</p>

中文翻译：

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传输方程的边界控制

<p style='text-indent:20px;'>本文考虑线性输运方程的两类边界控制问题。第一个显示在域的子域上的传输解决方案 <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline -formula> 可以从 <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline- 的传入边界条件精确控制公式>在适当的凸性假设下。这与通过应用独特的延拓属性通常为椭圆方程获得的唯一近似控制形成对比，我们证明了这一属性不适用于输运方程。我们还考虑了对来自输入条件的输出解的控制，这是一种类似于椭圆方程的 Dirichlet-to-Neumann 映射的传输概念。我们表明，对于传输方程中精心选择的系数，这种控制可能是不可能的。在这种情况下，通过 (Fredholm) 对偶，我们获得了与消失的输出条件兼容的非平凡输入条件的存在。</p>