Talk by Alessandro Duca

We consider the bilinear Schrödinger equation (BSE) $i\dd_t\psi=-\Delta\psi u(t)B\psi$ in the Hilbert space $L^2(\Gi,\C)$ when $\Gi$ is a graph type domain. The Laplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is a linear bounded symmetric operator in $L^2(\Gi,\C)$ and the function $u\in L^2((0,T),\R)$ is the control.

In the first part of the talk, we consider a compact graph $\Gi$ and we ensure the global exact controllability of the (BSE) when a specific spectral gap is verified. In the second part, we show how to validate such hypothesis in some examples of compact graphs $\Gi$. Finally, we discuss how to study the exact controllability of the (BSE) on infinite graphs despite the dispersive behaviour of the equation.