Null Controllability of Linear and Semilinear Nonlocal Heat Equations with an Additive Integral Kernel

Abstract

We consider a linear nonlocal heat equation in a bounded domain Ω ⊂ 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 timeindependent 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.2010 Mathematics Subject Classification. 35K58, 93B05, 93B07, 93C20.

Optimal control of linear non-local parabolic problems with an integral kernel

“…Moreover, in [23], analogous results have been obtained for a one-dimensional equation with a kernel in separated variables, by means of classical spectral analysis techniques. Later on, these mentioned results have been extended in [2], where the null-controllability is proved under weaker assumptions on the kernel and for both the linear and the semilinear case, by using a Carleman approach. In particular, the authors there proved the following result.…”

“…For the non-local diffusion models that we are going to consider in the present work, which will be introduced in the next section, controllability has been already treated in some recent paper (see, for instance, [2,9,19,23]). On the other hand, to the best of our knowledge, the analysis of optimal control problems was never addressed in this setting.…”

Controllability to trajectories of a Ladyzhenskaya model for a viscous incompressible fluid

“…Such terms can appear naturally in fluid mechanics to model the turbulence, but with more complicated models and we can also see such terms in biology, see for instance [15,Section 11.5]. Previous results have been obtained for parabolic systems with nonlocal spatial terms, see, for instance [1,8,9,13], etc. Let us note that in these references, the nonlocal spatial term is an integral term with a general kernel K (x, y) and here we only treat the case of a kernel of the form K = a ⊗ b, see (6) below.…”

Local null-controllability of a nonlocal semilinear heat equation

“…Indeed, even for linear equations, the by now classical Carleman estimates cannot handle in an easy way with the nonlocal terms. Let us mention a non exhaustive list of recent articles on the topic of controllability of nonlocal equations, see [FCLZ16] for linear heat equation with an analytic nonlocal spatial term, [LZ18] for linear systems, [BHS19] for linear and semilinear nonlocal heat equations, [FCLNHNnC19] for nonlocal nonlinear diffusion.…”