Null-Controllability of a Non-Local Heat Equation

TL;DR

This paper proves that a symmetric, square-integrable kernel ensures null-controllability of a non-local heat equation, extending previous conjectures and providing control cost estimates.

Contribution

It confirms a conjecture by showing that symmetry and L^2 integrability of the kernel suffice for null-controllability of the non-local heat equation.

Findings

Null-controllability holds under symmetric, L^2 kernel conditions.

The approach treats the non-local operator as a compact perturbation of the Laplacian.

Control cost estimates are derived using the perturbation bounds.

Abstract

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.