On the global approximate controllability in small time of semiclassical 1-D Schrödinger equations between two states with positive quantum densities

Abstract

In this paper, we study, in the semiclassical sense, the global approximate controllability in small time of the quantum density and quantum momentum of the 1-D semiclassical cubic Schrödinger equation with two controls between two states with positive quantum densities. We first control the asymptotic expansions of the zeroth and first order of the physical observables via Agrachev-Sarychev's method. Then we conclude the proof through techniques of semiclassical approximation of the nonlinear Schrödinger equation.

Small-time bilinear control of Schrödinger equations with application to rotating linear molecules

“…Saturation techniques have been introduced by Agrachev and Sarychev [1,2] to study the approximate controllability of 2D Navier-Stokes and Euler systems with additive controls, and extended to the 3D case in [23,24]. Other recent developments of these techniques are given, e.g., in [13] to study small-time controllability properties of semiclassical Schr ödinger equations, and in [15] to study local exact controllability of 1D Schr ödinger equations with Dirichlet boundary conditions.…”

Local exact controllability of the 1D nonlinear Schrödinger equation in the case of Dirichlet boundary conditions

“…Global controllability of the NLS equation (0.1) with κ = 0 is a challenging open problem. First results in this direction have been obtained recently by the authors [DN21] and by Coron et al [CXZ21], who consider approximate controllability between some particular states (in a semiclassical sense in the second reference). From the Main Theorem and the time reversibility of the Schrödinger equation it follows that global exact controllability will be established if one shows approximate controllability to the ground state φ 1 in the H 3 (0) -norm (see Theorem 3.2 in [Ner10] for the case κ = 0).…”