Small-time approximate controllability of the logarithmic Schr\''dinger equation

We consider Schr{ö}dinger equations with logarithmic nonlinearity and bilinear controls, posed on ${\mathbb{T}}^d$ or ${\mathbb{R}}^d$. We prove their small-time global $L^2$-approximate controllability. The proof consists in extending to this nonlinear framework the approach introduced by the first and third authors in \cite{beauchard-pozzoli2} to control the linear equation: it combines the small-time controllability of phases and gradient flows. Due to the nonlinearity, the required estimates are more difficult to establish than in the linear case. The proof here is inspired by WKB analysis. This is the first result of (small-time) global approximate controllability, for nonlinear Schr{ö}dinger equations, with bilinear controls.