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Communications on Pure and Applied Analysis (CPAA)
 

Persistent regional null contrillability for a class of degenerate parabolic equations

Pages: 607 - 635, Volume 3, Issue 4, December 2004

doi:10.3934/cpaa.2004.3.607       Abstract        Full Text (259.8K)       Related Articles

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)

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.

Keywords:  Degenerate parabolic equations, null controllability, observability inequalities
Mathematics Subject Classification:  35K65, 93B05, 93B07

Received: September 2003;      Revised: May 2004;      Published: September 2004.