Finite extinction and null controllability via delayed feedback non-local actions

We give sufficient conditions to have the finite extinction for all solutions of linear parabolic reaction-diffusion equations of the type ∂u ∂t − ∆u = −M(t)u(t − τ , x) (1) with a delay term τ > 0, on Ω, an open set of R N , M(t) is a bounded linear map on L p (Ω), u(t, x) satisfies a homogeneous Neumann or Dirichlet boundary condition. We apply this result to obtain distributed null controllability of the linear heat equation u t −∆u = v(t, x) by means of the delayed feedback term v(t, x) = −M(t)u(t − τ , x).