Small-time approximate controllability of the logarithmic Schrödinger equation

Univ. Rennes, CNRS, IRMAR, UMR 6625, 35000 Rennes, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 16 October 2025
Accepted: 3 April 2026

Abstract

We consider Schrödinger equations with logarithmic nonlinearity and bilinear controls, posed on T^d or ℝ^d. We prove their small-time global L²-approximate controllability. The proof consists in extending to this nonlinear framework the approach introduced by the first and third authors in [K. Beauchard and E. Pozzoli, Ann. Inst. H. Poincaré Anal. Non Linéaire (2025)] 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 Schrodinger equations, with bilinear controls.