Rapid stabilization of a degenerate parabolic equation using a backstepping approach: the case of a boundary control acting at the degeneracy

Pierre Lissy (1) , Claudia Moreno (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

The rapid exponential stabilization of a degenerate parabolic equation on a bounded interval with a left Dirichlet control is studied in this paper. Our approach introduces a backstepping transformation of Fredholm type, in order to force the solution of the closed-loop system to decay exponentially to zero with an arbitrary decay rate. The existence, continuity and invertibility of the transformation are obtained thanks to the study of Bessel functions of the first and the second kind, together with abstract results on Riesz bases and some specific properties satisfied by Fredholm operators.

Mots clés: Degenerate parabolic equation; backstepping; stabilization; Fredholm transform; Bessel functions

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Pierre Lissy, Claudia Moreno. Rapid stabilization of a degenerate parabolic equation using a backstepping approach: the case of a boundary control acting at the degeneracy. 2022. ⟨hal-03689515v2⟩