Persistent regional null contrillability for a class of degenerate parabolic equations
Pages: 607 - 635,
Volume 3, Issue 4,
Full Text (259.8K)
Piermarco Cannarsa - Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy (email)
Patrick Martinez - Laboratoire de Mathématiques MIP, UMR CNRS 5640, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31 062 Toulouse cedex 4, France (email)
Judith Vancostenoble - Laboratoire de Mathématiques MIP, UMR CNRS 5640, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31 062 Toulouse cedex 4, France (email)
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.
Keywords: Degenerate parabolic equations, null controllability, observability
Mathematics Subject Classification: 35K65, 93B05, 93B07
Received: September 2003;
Published: September 2004.