2024/10/03 by Beauchard, Karine, Pozzoli, Eugenio · 5 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2410.02383
We consider Schrödinger PDEs, posed on a boundaryless Riemannian manifold M, with bilinear control. We propose a new method to prove the global L2-approximate controllability. Contrarily to previous ones, it works in arbitrarily small time and does not require a discrete spectrum. This approach consists in controlling separately the radial part and the angular part of the wavefunction thanks to the control of the group \rm Diffc0(M) of diffeomorphisms of M and the control of phases, which refer to the possibility, for any initial state ψ0∈ L2(M,ℂ), diffeomorphism P∈ \rm Diffc0(M) and phase φ∈ L2(M,ℝ) to reach approximately the states (det DP)1/2(ψ0∘ P) and ei φψ0 . The control of the radial part uses the transitivity of the group action of \rm Diffc0(M) on positive densities proved by Moser. We develop this approach on two examples of Schrödinger equations, posed on \mathbbTd or ℝd, for which the small-time control of phases was recently proved. We prove that it implies the small-time control of flows of vector fields thanks to Lie bracket techniques. Combining this property with the simplicity of the group \rm Diffc0(M) proved by Thurston, we obtain the control of the group \rm Diffc0(M).