Null controllability for degenerate parabolic equations with internal control applied on a measurable subset

This paper investigates the null controllability of a class of two-dimensional linear parabolic equations with a boundary degeneracy on the domain Ω = 𝕋 × ( 0 , 1 ) , where the internal control acts on a space-time measurable subset of positive measure. Unlike conventional open control supports, a measurable control domain poses significant analytical challenges due to its lack of geometric and topological structures. To overcome this, we combine the method of separation of variables with sharp one-dimensional Carleman estimates that explicitly incorporate high-frequency parameters. This framework allows us to derive a refined Lebeau-Robbiano spectral inequality, which effectively bridges the gap from observability on open sets to general measurable subsets. Consequently, the exact null controllability with an 𝐿 ∞ -bounded control is established. This approach provides a unified analytical tool for proving controllability results for higher-dimensional degenerate parabolic equations on structured domains.