Null control of heat equations with analytic memory kernels

TL;DR

This work addresses memory-type null controllability for the heat equation with memory term $y_t - \int_0^t M(t-s)\Delta y(s)\,ds - b\Delta y = \chi\_{\omega(t)}u$ by analyzing how kernel regularity and moving control support $\omega(t)$ influence reachability of $y(T)=0$ under the memory constraint. The authors employ a reduction to heat-ODE coupled systems through memory variables $z_1(t)=\int_0^t M(t-s)y(s)\,ds$ (and higher orders), leveraging Carleman-based observability for the adjoint to obtain results. They show that polynomial kernels yield a finite cascade of ODEs and allow memory-type null controllability under moving-control coverage, while analytic kernels induce an infinite cascade, posing open problems for extending Carleman methods to this setting. The results delineate the boundary between tractable memory kernels and analytic ones for diffusion with moving controls and have implications for viscoelastic and memory-diffusion models, highlighting the role of geometric control in memory-enabled diffusion processes.

Abstract

We analyze the control properties of heat equations with memory terms. We recall previous results showing that if the moving support of the control covers the whole domain where heat diffuses, the system is null controllable when the memory kernel is polynomial. We formulate the problem of extending this result to the case of some more general memory kernels, in particular analytic ones.