Control of the Cauchy problem on Hilbert spaces: A global approach via symbol criteria

Let $A$ and $B$ be invariant linear operators with respect to a decomposition $\{H_{j}\}_{j\in \mathbb{N}}$ of a Hilbert space $\mathcal{H}$ in subspaces of finite dimension. We give necessary and sufficient conditions for the controllability of the Cauchy problem $$u_t=Au+Bv,\,\,u(0)=u_0,$$ in terms of the (global) matrix-valued symbols $\sigma_A$ and $\sigma_B$ of $A$ and $B,$ respectively, associated to the decomposition $\{H_{j}\}_{j\in \mathbb{N}}$. Then, we present some applications including the controllability of the Cauchy problem on compact manifolds for elliptic operators and the controllability of fractional diffusion models for H\"ormander sub-Laplacians on compact Lie groups. We also give conditions for the controllibility of wave and Schr\"odinger equations in these settings.Comment: 38 Page

Numerical Stackelberg--Nash Control for the Heat Equation

This paper deals with a strategy to solve numerically control problems of theStackelberg--Nash kind for heat equations with Dirichlet boundary conditions. We assume thatwe can act on the system through several controls, respecting an order and a hierarchy: a first con-trol (the leader) is assumed to choose the policy; then, a Nash equilibrium pair, determined by thechoice of the leader and corresponding to a noncooperative multiple-objective optimization strat-egy, is found (these are the followers). Our method relies on a formulation inspired by the work ofFursikov and Imanuvilov. More precisely, we minimize over the class of admissible null controls afunctional that involves weighted integrals of the state and the control, with weights that blow upat the final time. The use of the weights is crucial to ensure the existence of the controls and theassociated state in a reasonable space. We present several mixed formulations of the problems and,then, associated mixed finite element approximations that are easy to handle. In a final step, weexhibit some numerical experiments making use of the Freefem++ package

Weighted inequalities in Fluid mechanics and general relativity: Carleman estimates and cusped travelling waves

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 16-07-2019Esta tesis tiene embargado el acceso al texto completo hasta el 16-01-202

Control in moving interfaces and deep learning

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de Lectura: 14-05-2021This thesis has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex