2025/01/07 by Karine Beauchard, Beauchard, Karine, Frédéric Marbach +3
Engineering · Mathematics · #35Q41 #93B05 #93C20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2501.03882
openalex publication_date 2025/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the small-time local controllability (STLC) of a bilinear Schrödinger equation with Neumann boundary conditions near its ground state. We focus on the degenerate case where the linearized system is not controllable, necessitating a second-order analysis. We prove two complementary results. The negative result provides a new PDE instance of Sussmann's classical quadratic obstruction, corresponding to a non-vanishing Lie bracket. The positive result appears to be the first to establish STLC at the quadratic order for a physical PDE with a single scalar control. Both proofs rely on a Fourier-based approach, which is crucial because the integral kernel of the second-order term lacks the regularity required by standard integration-by-parts arguments. Along the way, we develop tools valid in a more general setting to analyze such quadratic forms. In particular, we prove results that allow for the multiplication of a kernel by a modulation function.