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Abstract
Motivated by physical models and the so-called Crocco equation,
we study the controllability properties of
a class of degenerate parabolic equations.
Due to degeneracy, classical null controllability results do not hold
for this problem in general.
First, we prove that we can drive the solution to rest
at time $T$ in a suitable subset of the space domain
(regional null controllability).
However, unlike for nondegenerate parabolic equations,
this property is no more automatically preserved with
time. Then, we prove that, given a time interval $(T,T')$,
we can control the equation up to $T'$ and remain at rest
during all the time interval $(T,T')$ on the same subset of the space domain
(persistent regional null controllability).
The proofs of these results are obtained via new observability
inequalities derived from classical Carleman estimates
by an appropriate use of cut-off functions.
With the same method, we also derive results of regional controllability
for a Crocco type linearized equation
and for the nondegenerate heat equation in unbounded domains.
Mathematics Subject Classification: 35K65, 93B05, 93B07.
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