Talk by Alessandro Duca
On April 6th, 2022, Dr. Alessandro Duca (Université Paris-Saclay) gave a talk about "Global exact controllability of the bilinear Schrödinger equation on graphs" as part of the research seminar Analysis of the FernUniversität in Hagen.
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.