Null-Controllability of a Non-Local Heat Equation

We consider the null-controllability of a non-local heat equation by interior $L^2(\Omega)$ controls. We confirm a conjecture of Lissy and Zuazua by showing that it is enough to assume that the kernel $k(x,\xi)$ is symmetric and $k(x,\xi)\in L^2(\Omega\times\Omega)$ in order to obtain the result. The result is obtained by treating the non-local linear operator $K:L^2(\Omega)\rightarrow L^2(\Omega)$ as a compact and bounded perturbation of the Dirichlet-Laplacian, $A$, which leads to useful bounds on the semigroup generated by $(A+K,\mathcal{D}(A))$. Using this approach, we are also able to estimate the cost of control.