Quadratic obstructions to Small-Time Local Controllability for the multi-input bilinear Schr\"odinger equation

TL;DR

This paper demonstrates that quadratic obstructions can prevent small-time local controllability in multi-input bilinear Schrödinger equations, extending known single-input results to PDEs and providing a new analytical framework.

Contribution

It introduces the first example of multi-input quadratic obstructions for PDEs, expanding the understanding of controllability limitations in bilinear Schrödinger equations.

Findings

Quadratic terms can obstruct controllability in multi-input PDEs.

Extension of single-input quadratic obstruction results to multi-input PDEs.

Development of a functional framework for analyzing these obstructions.

Abstract

We investigate the small-time local controllability (STLC) near the ground state of a bilinear Schr\"odinger 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 Schr\"odinger equation.