Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 115, pp. 1-13.

Title: Optimal bilinear control for Gross-Pitaevskii equations with singular potentials

Authors: Kai Wang (Lanzhou  Univ., Lanzhou, China)
         Dun Zhao (Lanzhou  Univ., Lanzhou, China)

Abstract:
 We study the optimal bilinear control problem of the generalized
 Gross-Pitaevskii equation
 $$
 i\partial_{t}u=-\Delta u+U(x)u+\phi(t)\frac{1}{|x|^{\alpha}}u
  +\lambda|u|^{2\sigma}u,\quad  x\in \mathbb{R}^3,
 $$
 where U(x) is the given external potential, $\phi(t)$ is the control function.
 The existence of an optimal control and the optimality condition are presented
 for suitable $\alpha$ and $\sigma$. In particular, when $1\leq\alpha<3/2$,
 the Frechet-differentiability of the objective functional is proved for
 two cases:
 (i) $\lambda<0$, $0<\sigma<2/3$;
 (ii) $\lambda>0$, $0<\sigma<2$. 
 Comparing with the previous studies in [6],  the results fill the gap for 
 $\sigma \in (0,1/2)$.

Submitted February 10, 2019. Published October 13, 2019.
Math Subject Classifications: 35Q55, 49J20.
Key Words: Optimal bilinear control; Gross-Pitaevskii equation; objective functional; 
           Frechet-differentiability; optimal condition.