发表机构
Keio University; University of Luxembourg（庆应义塾大学; 卢森堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文精确确定了输运-扩散方程在粘性消失极限下的均匀零可控时间：正速度时为 $2L/M$，负速度时为 $(2+2\sqrt{2})L/|M|$，否定了 Coron--Guerrero 猜想，并通过伴随解构造和模型空间方法证明了下界与上界。
AI 中文摘要
我们研究在粘性消失极限下，$[0,L]$ 上的输运-扩散方程 $u_t + M u_x = \varepsilon u_{xx}$ 在 $x=0$ 处具有 Dirichlet 边界控制的均匀零可控性。我们的结果涉及均匀零可控时间 $T_{\mathrm{unif}}$，定义为当 $\varepsilon \to 0$ 时零可控成本保持有界的时间的下确界。我们精确确定了输运速度两种符号下的该时间：正速度情形 $M>0$ 时 $T_{\mathrm{unif}} = 2 L / M$，负速度情形 $M<0$ 时 $T_{\mathrm{unif}} = (2 + 2 \sqrt{2}) L / |M|$。正速度结果否定了 Coron 和 Guerrero 提出的猜想 $T_{\mathrm{unif}}=L/M$。对于负速度，Lissy 已经否定了猜想阈值。我们的结果确定了精确阈值。我们通过构造违反相应阈值以下均匀可观测性的伴随解来建立下界。在正速度情形，构造利用了与伴随谱生成的指数族相关的 Blaschke 乘积和模型空间的渐近结构。相同的模型空间结构被用来将 $M$ 两种符号的上界问题归结为无限时间可观测性不等式，这些不等式通过边界到内部映射的表示及其 Hilbert--Schmidt 范数的估计来证明。
英文摘要
We study the uniform null-controllability, in the vanishing viscosity limit, of the transport--diffusion equation $u_t + M u_x = \varepsilon u_{xx}$ on $[0,L]$ with a Dirichlet boundary control at $x=0$. Our results concern the uniform null-controllability time $T_{\mathrm{unif}}$, defined as the infimum of the times for which the null-controllability cost remains bounded as $\varepsilon \to 0$. We determine this time exactly for both signs of the transport velocity: $T_{\mathrm{unif}} = 2 L / M$ in the positive-speed case $M>0$, and $T_{\mathrm{unif}} = (2 + 2 \sqrt{2}) L / |M|$ in the negative-speed case $M<0$. The positive-speed result disproves the conjecture $T_{\mathrm{unif}}=L/M$ suggested by Coron and Guerrero. For negative speed, the conjectured threshold had already been disproved by Lissy. Our result determines the exact threshold. We establish the lower bounds by constructing adjoint solutions that violate uniform observability below the respective thresholds. In the positive-speed case, the construction exploits the asymptotic structure of the Blaschke products and model spaces associated with the exponential family generated by the adjoint spectrum. The same model-space structure is used to reduce the upper-bound problems for both signs of $M$ to infinite-time observability inequalities, which are proved through a representation of the boundary-to-interior map and estimates of its Hilbert--Schmidt norm.
