Null controllability of linear and semilinear nonlocal heat equations with integral kernel

TL;DR

This paper extends null controllability results for linear and semilinear nonlocal heat equations with integral kernels, relaxing previous assumptions and providing explicit controllability cost estimates.

Contribution

It generalizes null controllability to broader classes of kernels and derives explicit cost estimates, enabling application to semilinear models.

Findings

Null controllability achieved for more general kernels.

Explicit estimates on controllability cost derived.

Extension to certain semilinear models demonstrated.

Abstract

We consider a linear nonlocal heat equation in a bounded domain $\Omega\subset\mathbb{R}^d$ with Dirichlet boundary conditions. The non-locality is given by the presence of an integral kernel. We analyze the problem of controllability when the control acts on an open subset of the domain. It is by now known that the system is null-controllable when the kernel is time-independent and analytic or, in the one-dimensional case, in separated variables. In this paper, we relax this assumption and we extend the result to a more general class of kernels. Moreover, we get explicit estimates on the cost of null-controllability that allow us to extend the result to some semilinear models.