Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tunneling estimates and approximate controllability for hypoelliptic
  equations   This article is concerned with quantitative unique continuation estimates for
equations involving a "sum of squares" operator $\mathcal{L}$ on a compact
manifold $\mathcal{M}$ assuming: $(i)$ the Chow-Rashevski-H\"ormander condition
ensuring the hypoellipticity of $\mathcal{L}$, and $(ii)$ the analyticity of
$\mathcal{M}$ and the coefficients of $\mathcal{L}$.
  The first result is the tunneling estimate $\|\varphi\|_{L^2(\omega)} \geq
Ce^{- \lambda^{\frac{k}{2}}}$ for normalized eigenfunctions $\varphi$ of
$\mathcal{L}$ from a nonempty open set $\omega\subset \mathcal{M}$, where $k$
is the hypoellipticity index of $\mathcal{L}$ and $\lambda$ the eigenvalue.
  The main result is a stability estimate for solutions to the hypoelliptic
wave equation $(\partial_t^2+\mathcal{L})u=0$: for $T>2 \sup_{x \in
\mathcal{M}}(dist(x,\omega))$ (here, $dist$ is the sub-Riemannian distance),
the observation of the solution on $(0,T)\times \omega$ determines the data.
The constant involved in the estimate is $Ce^{c\Lambda^k}$ where $\Lambda$ is
the typical frequency of the data.
  We then prove the approximate controllability of the hypoelliptic heat
equation $(\partial_t+\mathcal{L})v=1_\omega f$ in any time, with appropriate
(exponential) cost, depending on $k$. In case $k=2$ (Grushin, Heisenberg...),
we further show approximate controllability to trajectories with polynomial
cost in large time.
  We also explain how the analyticity assumption can be relaxed, and a boundary
$\partial \mathcal{M}$ can be added in some situations.
  Most results turn out to be optimal on a family of Grushin-type operators.
  The main proof relies on the general strategy developed by the authors in
arxiv:1506.04254.

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The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
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The categories of this paper are: stat