论文标题
层次波方程永远无法观察到
Subelliptic wave equations are never observable
论文作者
论文摘要
众所周知，椭圆波方程的可观察性（并通过二元性，可控性），即带有riemannian laplacian，随着时间的推移，$ t_0 $几乎等同于几何控制条件（GCC），该条件（GCC）规定，任何地球射线都符合时间内的控制时间$ t_0 $ t_0 $ $ t_0 $。我们表明，在下层次设置中，GCC从未得到验证，并且在有限的时间内永远无法观察到下去的波动方程。更准确地说，给定任何亚段落拉普拉斯$δ= - \ sum_ {i = 1}^m x_i^*x_i^*x_i $在歧管上$ m $上，以及任何可测量的子集$ω\ subset m $，使得$ m \ m \backslashΩ$在其内部$ q $中包含$ q $， \ text {span}（x_1，\ ldots，x_m）$对于某些$ 1 \ leq i，j \ leq m $，我们表明，对于任何$ t_0> 0 $，带有subelliptic laplacian $δ$的波动方程在$ p_0 $ t_0 $上都无法观察到$ω$。该证明是基于集中在大地测量学上的波方程序列的构建（对于相关的亚riemannian距离），将很长一段时间用于$ m \ backslashω$。作为同行，我们证明了海森堡组的波动方程可观察性的积极结果，在海森堡组中，观察集是相位空间的精心选择的一部分。
It is well-known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the Geometric Control Condition (GCC), which stipulates that any geodesic ray meets the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $Δ=-\sum_{i=1}^m X_i^*X_i$ on a manifold $M$, and any measurable subset $ω\subset M$ such that $M\backslash ω$ contains in its interior a point $q$ with $[X_i,X_j](q)\notin \text{Span}(X_1,\ldots,X_m)$ for some $1\leq i,j\leq m$, we show that for any $T_0>0$, the wave equation with subelliptic Laplacian $Δ$ is not observable on $ω$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on geodesics (for the associated sub-Riemannian distance) spending a long time in $M\backslash ω$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space.
