Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability

TL;DR

This paper establishes the global exact controllability of bilinear Schrödinger equations on compact graphs and introduces a new concept called energetic controllability, with applications to specific graph structures like star graphs.

Contribution

The work presents a novel technique for proving global exact controllability of bilinear quantum systems on compact graphs and introduces energetic controllability as a new, weaker controllability notion.

Findings

Proves global exact controllability in certain Sobolev spaces.

Introduces and formalizes energetic controllability.

Applies results to star graph configurations.

Abstract

The aim of this work is to study the controllability of the bilinear Schr\"odinger equation on compact graphs. In particular, we consider the equation (BSE) $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$, with $\mathscr{G}$ being a compact graph. The Laplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We provide a new technique leading to the global exact controllability of the (BSE) in $D(|\Delta|^{s/2})$ with $s\geq 3$. Afterwards, we introduce the "energetic controllability", a weaker notion of controllability useful when the global exact controllability fails. In conclusion, we develop some applications of the main results involving for instance star graphs.