Local controllability of the bilinear 1D Schr{ö}dinger equation with simultaneous estimates

Abstract

We consider the 1D linear Schrödinger equation, on a bounded interval, with Dirichlet boundary conditions and bilinear scalar control. The small-time local exact controllability around the ground state was proved in [5], under an appropriate nondegeneracy assumption. Here, we work under a weaker nondegeneracy assumption and we prove the small-time local exact controllability in projection, around the ground state, with estimates on the control (depending linearly on the target) simultaneously in several spaces. These estimates are obtained at the level of the linearized system, thanks to a new result about trigonometric moment problems. Then, they are transported to the nonlinear system by the inverse mapping theorem, thanks to appropriate estimates of the error between the nonlinear and the linearized dynamics.

Quadratic behaviors of the 1D linear Schr{ö}dinger equation with bilinear control

“…Notice that the right-hand side of the equality is indeed in H 1 0 (0, 1) thanks to the smoothing effect stated below in Lemma 4.4, which was highlighted in [5] and later used in [11].…”

“…Error estimates for the auxiliary system. We need sharp error estimates to prove that the cubic remainder of the expansion (11) can be neglected in front of the drift u 1 2 L 2 . Therefore, classical error estimates involving the L 2 -norm of the control u are not enough.…”

“…In this section, we recall the result about the well-posedness of the following Cauchy problem, given in [5, Proposition 2] and later extended in [11],…”

“…We will also always take ϕ 1 as the initial condition of (1). Besides, as highlighted in [11], the solution of ( 28) and more precisely its regularity, relies strongly on the control and the dipolar moment. However, for sake of simplicity, in the following, we will not mention this dependency.…”

Small-time local controllability of the bilinear Schrödinger equation, despite a quadratic obstruction, thanks to a cubic term

“…Given a target, we first realize the expected motion along the lost direction by using control variations for which the cubic term dominates the quadratic one. Then, we correct the other components exactly, by using the STLC in projection result of [15], with simultaneous estimates of weak norms of the control. These estimates ensure that the new error along the lost direction is negligible, and we conclude with the Brouwer fixed point theorem.…”