次椭圆波动方程永远不可观测
Subelliptic wave equations are never observable
AI总结：
该研究证明次椭圆波动方程在有限时间内不可观测，并通过构造沿次黎曼测地线集中且避开观测集的解序列给出论证，同时在 Heisenberg 群中获得选取相空间区域的正面可观测结果。
摘要：
众所周知，椭圆波动方程（即带有黎曼 Laplace 算子的波动方程）在时间 $T_0$ 内的可观测性（并通过对偶性，即可控性）几乎等价于几何控制条件（GCC），该条件规定任何测地射线都在时间 $T_0$ 内遇到控制集。我们证明，在次椭圆情形下，GCC 永远不会被满足，并且次椭圆波动方程在有限时间内永远不可观测。更准确地说，给定流形 $M$ 上的任意次椭圆 Laplace 算子 $Δ=-\sum_{i=1}^m X_i^*X_i$，以及任意可测子集 $ω\subset M$，且 $M\backslash ω$ 的内部包含一点 $q$，使得对于某些 $1\leq i,j\leq m$ 有 $[X_i,X_j](q)\notin \text{Span}(X_1,\ldots,X_m)$，我们证明：对于任意 $T_0>0$，带有次椭圆 Laplace 算子 $Δ$ 的波动方程在时间 $T_0$ 内于 $ω$ 上不可观测。证明基于构造一系列波动方程解，这些解集中在（相关次黎曼距离意义下的）测地线上，并在 $M\backslash ω$ 中停留很长时间。作为对应结果，我们证明了 Heisenberg 群中波动方程的一个正面可观测性结果，其中观测集是相空间中精心选取的一部分。
英文摘要：
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.
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