Global Exact Controllability of Bilinear Quantum Systems on Compact Graphs and Energetic Controllability

Abstract

The aim of this work is to study the controllability of the bilinear Schrödinger 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 $(BSE)$ 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.