Some new results related to the null controllability of the 1-d heat equation

We address three null controllability problems related to the $1-d$ heat equation. First we show that the $1-d$ heat equation with a rapidly oscillating density is uniformly null controllable as the period of the density tends to zero. We also prove that the same result holds for the finite-difference semi-discretization in space of the constant coefficient heat equation as the step size tends to zero. Finally, we prove that the null controllability of the constant coefficient heat equation can be obtained as limit of null controllability properties for singularly perturbed dissipative wave equations. The proofs combine results on sums of real exponentials, Carleman’s inequalities for heat equations and sideways energy estimates for wave equations