U. Biccari, V. Hernández-Santamaría Null controllability of a nonlocal heat equation with integral kernel, DOI: 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 a…

#### Null-controllability properties of a fractional wave equation with a memory term

U. Biccari, M. Warna Null-controllability properties of a fractional wave equation with a memory term Abstract: We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require…

#### Controllability of the one-dimensional fractional heat equation under positivity constraints

U. Biccari, M. Warna, E. Zuazua Internal observability for coupled systems of linear partial differential equations Abstract: In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian $(-\Delta)^s$ ($0

#### Null-controllability properties of the wave equation with a second order memory term

U. Biccari, S. Micu Null-controllability properties of the wave equation with a second order memory term,doi.org/10.1016/j.jde.2019.02.009 Abstract: We study the internal controllability of a wave equation with memory in the principal part, defined on the one-dimensional torus $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$. We assume that the control is acting on an open subset $\omega(t)\subset\mathbb{T}$, which is moving with a…

#### Dynamics and control for multi-agent networked systems: a finite difference approach

U. Biccari, D. Ko, E. Zuazua Dynamics and control for multi-agent networked systems: a finite difference approach Abstract: We analyze the dynamics of multi-agent collective behavior models and their 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…

#### Propagation of one and two/dimensional discrete waves under finite difference approximation

U. Biccari, A. Marica, E. Zuazua Propagation of one and two/dimensional discrete waves under finite difference approximation, DOI: Abstract: We analyze the propagation properties of the numerical versions of one and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both at the continuous…

#### Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential

U. Biccari Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential, Mathematical Control and related fields, DOI: 10.3934/mcrf.2019011 Abstract: We analyse controllability properties for the one-dimensional heat equation with singular inverse-square potential $u_t-u_{xx}-\frac{\mu}{x^2}u=0\,\,\,(x,t)\in(0,1)\times(0,T)$. For any $\mu

#### The Poisson equation from non-local to local

U. Biccari, V. Hernández-Santamaría The Poisson equation from non-local to local, Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 145, pp. 1-13. DOI: arXiv:1801.09470 Abstract: We analyze the limit behavior as $s\to 1^-$ of the solution to the fractional Poisson equation $\fl{s}{u_s}=f_s$, $x\in\Omega$ with homogeneous Dirichlet boundary conditions $u_s\equiv 0$, $x\in\Omega^c$. We show that…

#### Null controllability of a nonlocal heat equation with integral kernel

U. Biccari, V. Hernández-Santamaría Null controllability of a nonlocal heat equation with integral kernel, DOI: 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 a…

#### WKB expansion for a fractional Schrödinger equation with applications to controllability

U. Biccari, A. B. Aceves WKB expansion for a fractional Schrödinger equation with applications to controllability, DOI: Abstract: This paper is devoted to the construction of localized quasi-solutions in general optics for a one-dimensional nonlocal Schrödinger equation involving the fractional Laplace operator. A suitable ansatz for the solutions to the problem is obtained adapting a…