This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator
L
\mathcal {L}
on a compact manifold
M
\mathcal {M}
assuming:
(
i
)
(i)
the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of
L
\mathcal {L}
, and
(
i
i
)
(ii)
the analyticity of
M
\mathcal {M}
and the coefficients of
L
\mathcal {L}
.
The first result is the tunneling estimate
‖
φ
‖
L
2
(
ω
)
≥
C
e
−
c
λ
k
2
\|\varphi \|_{L^2(\omega )} \geq Ce^{- c\lambda ^{\frac {k}{2}}}
for normalized eigenfunctions
φ
\varphi
of
L
\mathcal {L}
from a nonempty open set
ω
⊂
M
\omega \subset \mathcal {M}
, where
k
k
is the hypoellipticity index of
L
\mathcal {L}
and
λ
\lambda
the eigenvalue.
The main result is a stability estimate for solutions to the hypoelliptic wave equation
(
∂
t
2
+
L
)
u
=
0
(\partial _t^2+\mathcal {L})u=0
: for
T
>
2
sup
x
∈
M
(
d
i
s
t
(
x
,
ω
)
)
T>2 \sup _{x \in \mathcal {M}}(dist(x,\omega ))
(here,
d
i
s
t
dist
is the sub-Riemannian distance), the observation of the solution on
(
0
,
T
)
×
ω
(0,T)\times \omega
determines the data. The constant involved in the estimate is
C
e
c
Λ
k
Ce^{c\Lambda ^k}
where
Λ
\Lambda
is the typical frequency of the data.
We then prove the approximate controllability of the hypoelliptic heat equation
(
∂
t
+
L
)
v
=
1
ω
f
(\partial _t+\mathcal {L})v=\mathbb {1}_\omega f
in any time, with appropriate (exponential) cost, depending on
k
k
. In case
k
=
2
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
∂
M
\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 to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).