Carleman estimates for degenerate parabolic operators with applications to null controllability
Abstract
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.