论文标题
弱耗散热的方程式的无效控制性
Null-controllability for weakly dissipative heat-like equations
论文作者
论文摘要
我们研究了在整个欧几里得空间上构成的热式方程式的无数可控性能。这些演变方程与表单$ρ（\ vert d_x \ vert）$的傅立叶乘数相关联，其中$ρ\ colon [0，+\ infty）\ rightArrow \ rightArrow \ mathbb c $是一个可测量的函数，因此$ \ rec \ re phe是从下面界限的。我们考虑``弱耗散''情况，一个典型的示例是由与乘数$ρ（ξ）=ξ^s $相关的分数方程给出的，在（0,1）$中，很少有结果。我们确定了对控制支持的足够条件和必要条件，以持有零控制。更确切地说，我们证明，在任何积极的时间内，这些方程在所有尺度上都足够厚的控制载体中都是无效控制的。在乘数$ρ$的假设下，特别是假设$ρ（ξ）= O（ξ）$，我们还证明了零控制性意味着控制支持在所有规模上都是较厚的，并且具有倍增器$ρ$的厚度比的明确下降范围。最后，使用Smith-Volterra-Cantor集合，我们提供了满足这些必要或充分条件的非平凡控制支持的示例。
We study the null-controllability properties of heat-like equations posed on the whole Euclidean space $\mathbb R^n$. These evolution equations are associated with Fourier multipliers of the form $ρ(\vert D_x\vert)$, where $ρ\colon[0,+\infty)\rightarrow\mathbb C$ is a measurable function such that $\Reρ$ is bounded from below. We consider the ``weakly dissipative'' case, a typical example of which is given by the fractional heat equations associated with the multipliers $ρ(ξ) = ξ^s$ in the regime $s\in(0,1)$, for which very few results exist. We identify sufficient conditions and necessary conditions on the control supports for the null-controllability to hold. More precisely, we prove that these equations are null-controllable in any positive time from control supports which are sufficiently thick at all scales. Under assumptions on the multiplier $ρ$, in particular assuming that $ρ(ξ) = o(ξ)$, we also prove that the null-controllability implies that the control support is thick at all scales, with an explicit lower bound of the thickness ratio in terms of the multiplier $ρ$. Finally, using Smith-Volterra-Cantor sets, we provide examples of non-trivial control supports that satisfy these necessary or sufficient conditions.
