Umberto Biccari is a PhD. Currently he holds a Postdoctoral position at the ERC Advanced Grant project DyCon under the supervision of Prof. Enrique Zuazua (UAM and DeustoTech). In the past, he collaborated within the ERC research project NUMERIWAVES. His current research interests are related to the analysis of Partial Differential Equations, in particular from the point of view of control theory. During the years of his PhD he has been concerned with the study of controllability properties of hyperbolic (waves), parabolic (heat) and dispersive (Schrödinger) PDEs, involving non-local terms, singular inverse-square potentials, variable degenerate coefficients or dynamical boundary conditions. At the moment, he is getting interested in non-local transport problems, derived from models of collection behaviour.

Simultaneous control of parameter-depending systems using stochastic optimization algorithms

The inverse design of hyperbolic transport equations can be addressed by using gradient-adjoint methodologies. Recently, Morales-Hernandez and Zuazua [1] investigated the convenience of using low order numerical schemes for the adjoint resolution in the gradient-adjoint method. They focused on hyperbolic transport scalar equations with an heterogeneous time-independent vector field.

In this tutorial, we show the simulation of heat fractional equation

We analyze the propagation properties of the numerical versions of one and two-dimensional wave equations, semi-discretized in space by finite difference schemes, and we illustrate that numerical solutions may have unexpected behaviours with respect to the analytic ones.

We describe a FE method for the approximation of the one-dimensional fractional Laplacian $(-d_x^2)^s$ on a uniform mesh discretizing the symmetric interval $(-L,L)$, $L>0$.

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 < s < 1 $) on the interval $(-1,1)$. We prove the existence of a minimal (strictly positive) time $T_{\rm min}$ such that the fractional heat dynamics can be controlled from any initial datum in $L^2(-1,1)$ to a positive trajectory through the action of a positive control, when $s>1/2$. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical results.

Theoretical and numerical analysis of the propagation of the solutions for a Schrödinger equation with fractional Laplacian, with application to the study of controllability properties.

Design of a LQR controller for the stabilization of a fractional reaction diffusion equation