Optimal Bilinear Control of Nonlinear Hartree Equations with Singular Potentials

Optimal nonlinearity control of Schrödinger equation

“…In comparison, the coherent manipulation of quantum systems via external potentials corresponding to the linear control where f (u) denotes the nonlinearity term, V (x) is a fixed external potential, and W (x, t) corresponds to the controller. A lot of works have been carried out on this problem, we refer to [1,12,13,14,15,16,17,19,28] for some related studies. However, for the nonlinearity control problem, although it has been studied in physics literature (see [6,7,20,21,22,24] and the references therein), to our knowledge, a mathematical discussion is still lacking.…”

Optimal bilinear control of the logarithmic Schrödinger equation

“…with g(u) = |u| 2 u. Thereafter, the case of g(u) = |u| 𝛼 u with 𝛼 ≥ 1 was considered in Feng and Zhao, 23 the case of Hartree-type nonlinearity g(u) = (W * |u| 2 )u was treated with in Feng and Wang, 26 and the method was generalized to the weakly coupled nonlinear Schrödinger system. 27 Besides the bilinear control based on external potential, the nonlinearity control based on Feshbach resonance management was studied in Wang et al 28 However, all of the works mentioned above depend heavily on the local Lipschitz (or Hölder) continuity of the solution u(𝜙) with respect to the control 𝜙.…”

“…$$ with $$$ g(u)={\left|u\right|}^2u $$$. Thereafter, the case of $$$ g(u)={\left|u\right|}^{\alpha }u $$$ with $$$ \alpha \ge 1 $$$ was considered in Feng and Zhao, 23 the case of Hartree‐type nonlinearity $$$ g(u)=\left(W\ast {\left|u\right|}^2\right)u $$$ was treated with in Feng and Wang, 26 and the method was generalized to the weakly coupled nonlinear Schrödinger system 27 . Besides the bilinear control based on external potential, the nonlinearity control based on Feshbach resonance management was studied in Wang et al 28 However, all of the works mentioned above depend heavily on the local Lipschitz (or Hölder) continuity of the solution $$$ u\left(\phi \right) $$$ with respect to the control $$$ \phi $$$.…”