Optimal bilinear control of the coupled nonlinear Schrödinger system

Optimal bilinear control of the logarithmic Schrödinger equation

“…However, when (𝛾, 𝜌) ≠ (∞, 2), we cannot get the uniformly boundedness of ||u 𝜀 || L 𝛾 t Σ 1,𝜌 (0,T) . Indeed, similar to the case of the power-law nonlinearities, see Wang et al, 27,28 we can deduce from Strichartz's estimates and Lemma 2.2 that M is depended by 𝜌, 𝜀, and ||u 𝜀 || L ∞ t Σ(0,T) .…”

“…$$ However, when $$$ \left(\gamma, \rho \right)\ne \left(\infty, 2\right) $$$, we cannot get the uniformly boundedness of $$$ {\left\Vert {u}_{\varepsilon}\right\Vert}_{L_t^{\gamma }{\Sigma}^{1,\rho}\left(0,T\right)} $$$. Indeed, similar to the case of the power‐law nonlinearities, see Wang et al, 27,28 we can deduce from Strichartz's estimates and Lemma 2.2 that $$$ M $$$ is depended by $$$ \rho $$$, $$$ \varepsilon $$$, and $$$ {\left\Vert {u}_{\varepsilon}\right\Vert}_{L_t^{\infty}\Sigma \left(0,T\right)} $$$.…”

“…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 $$$.…”