Quadratic behaviors of the 1D linear Schrödinger equation with bilinear control

Small-time local controllability of the bilinear Schrödinger equation with a nonlinear competition

“…The contribution of the quadratic term, along a lost direction, can be a signed quadratic form, that induces a drift of the solution along this direction, preventing controllability. In the small time asymptotic, this always happens in finite dimension [7] and is also observed in infinite dimension [9,13,21,22] (without being systematic [8]). For these obstructions to hold, the control needs to be small in a norm adapted to the Lie bracket along which the system drifts; thus the small-time local controllability was still open without this smallness assumption, and this is the problem under study in this article.…”

“…Quadratic behavior. In [13], the author proved that when the condition (1.14) does not hold, and more precisely, when for some n ≥ 2 (resp. n = 1)…”

“…• in the asymptotic of controls small in H 3 0 , Gagliardo-Nirenberg inequalities prove that the quadratic term prevails on the cubic term (and on the higher-order terms), and thus one can deny H 3 0 -STLC as done in [13]. The reader can refer to the toy model in Section 2.3 for more details on this competition between the quadratic and cubic terms.…”

“…We consider a dipolar moment µ for which the linearized system loses one direction, on which appears a competition between a quadratic and a cubic term. It is known [13] that the quadratic term prevents small-time local controllability in a certain regime (in which the cubic term is negligible compared to the quadratic term). In a different regime (in which the quadratic term is negligible compared to the cubic term), we prove that small-time local controllability holds.…”

“…Thus, the quadratic term prevails on the cubic term in (2.3) when controls are small in H 3 . This allows us to deny H 3 -STLC for (2.2) (see for instance [7,13] for more details on quadratic obstructions). Nonetheless, using the strategy described above, let us prove that (2.2) is H 2 0 -STLC.…”