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The aim of this work is to study the controllability of the bilinearSchr\"odinger equation on compact graphs. In particular, we consider theequation (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. TheLaplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is abounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. Weprovide 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 theglobal exact controllability fails. In conclusion, we develop some applicationsof the main results involving for instance star graphs.

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