Carleman estimates for degenerate parabolic operators with applications to null controllability

F. Alabau-Boussouira (1) , P. Cannarsa (2) , G. Fragnelli (3)

1 LMAM - Laboratoire de Mathématiques et Applications de Metz
2 DIPMAT - Dipartimento di Matematica [Roma II]
3 UNISI - Università degli Studi di Siena = University of Siena

We prove an estimate of Carleman type for the one dimensional heat equation ut − (a(x)ux )x + c(t, x)u = h(t, x), (t, x) ∈ (0, T ) × (0, 1), where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation ut − (a(x)ux )x + f (t,x,u) = h(t, x)χω(x), where (t, x) ∈ (0, T ) × (0, 1), ω = (α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u.

Mots clés: Degenerate parabolic equations; mill controllability; observability; Carleman estimates; Hardy inequality

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

F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli. Carleman estimates for degenerate parabolic operators with applications to null controllability. Journal of Evolution Equations, 2006, 6 (2), pp.161-204. ⟨10.1007/s00028-006-0222-6⟩. ⟨hal-03895117⟩