Quadratic obstructions to Small-Time Local Controllability for the multi-input bilinear Schrödinger equation
Théo Gherdaoui
TL;DR
The paper analyzes small-time local controllability of the multi-input bilinear Schrödinger equation around the ground state in a 1D quantum well, showing that quadratic terms create obstructions when the linearized system is missing a direction. By developing a second-order power-series expansion, deriving a Magnus-type representation, and establishing sharp error and interpolation estimates, it identifies a definite quadratic drift governed by the scalar series $\gamma_{2k-1}^{\ell,L}$. Under regularity and non-degeneracy hypotheses $\mathbf{(H)_{reg}}$, $\mathbf{(H)_{lin}}$, $\mathbf{(H)_{conv}}$, $\mathbf{(H)_{null}}$, and $\mathbf{(H)_{pos}}$, the authors prove that STLC fails in $H^{2k-3}$ (or in $W^{-1,\infty}$ when $k=1$), providing the first multi-input quadratic obstructions for PDEs. These results extend finite-dimensional obstructions to the PDE setting, offering a functional framework and precise conditions under which the drift prevents exact small-time controllability and informs the limits of linearization-based approaches in multi-input quantum control.
Abstract
We investigate the small-time local controllability (STLC) near the ground state of a bilinear Schrödinger equation when the linearized system is not controllable. It is well known that, for single-input systems, quadratic terms in the state expansion can then lead to obstructions to the STLC of the nonlinear system. In this work, we extend this phenomenon to the multi-input setting, presenting the first example of multi-input quadratic obstructions for PDEs. Our results build upon our previous study of such obstructions for ODEs and provide a functional framework for analyzing them in the bilinear Schrödinger equation.