Abstract. We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian (−∆) s on an arbitrary bounded open set Ω ⊂ R N . For 1 < p < 2, we obtain regularity in the Besov space B 2s p,2,loc (Ω), while for 2 ≤ p < ∞ we show that the solutions belong to W 2s,p loc (Ω). The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.Dedicated to Ireneo Peral on the occasion of his 70th birthday: Gracias Ireneo por tantos años de amistad y ejemplo.

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 $s&gt;1/2$ 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.

We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the non-local transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the non-local transport model can be obtained by a suitable averaging of the diffusive one.We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered.Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time-horizon and the size of the controls.2010 Mathematics Subject Classification. 34D20, 35B36, 65M06, 92D25, 93B05.

This article is devoted to the analysis of control properties for a heat equation with singular potential µ/δ 2 , defined on a bounded C 2 domain Ω ⊂ R N , where δ is the distance to the boundary function. More precisely, we show that for any µ ≤ 1/4 the system is exactly null controllable using a distributed control located in any open subset of Ω, while for µ > 1/4 there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate.

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.