Controllability of the Heat Equation with a Control Acting on a Measurable Set

The paper deals with the controllability of a heat equation.It is well-known that the heat equation yt - △y =uxE in (0,T) × Ω with homogeneous Dirichlet boundary conditions is null controllable for any T ＞ 0 and any open nonempty subset E of Ω.In this note,the author studies the case that E is an arbitrary measurable set with positive measure.