We analyse the controllability problem for a one-dimensional heat equation involving the fractional Laplacian $(-d_x^{\,2})^{s}$ on the interval $(-1,1)$. Using classical results and techniques, we show that, acting from an open subset $\omega \subset (-1,1)$, the problem is null-controllable for $s>1/2$ and that for $s\leqslant 1/2$ we only have approximate controllability. Moreover, we deal with the numerical computation of the control employing the penalized Hilbert Uniqueness Method and a finite element scheme for the approximation of the solution to the corresponding elliptic equation. We present several experiments confirming the expected controllability properties.

In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems.

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.