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

TL;DR

This paper establishes criteria for controlling the solutions of certain linear PDEs on Hilbert spaces using symbol analysis, with applications to manifolds, Lie groups, and quantum equations.

Contribution

It provides necessary and sufficient symbol-based conditions for controllability of linear PDEs on Hilbert spaces, extending to various geometric and quantum contexts.

Findings

Derived global matrix-valued symbol criteria for controllability.

Applied criteria to elliptic operators on manifolds and sub-Laplacians on Lie groups.

Established controllability conditions for wave and Schrödinger equations in these settings.

Abstract

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.