Electron. J. Differential Equations, Vol. 2019 (2019), No. 115, pp. 1-13.

Optimal bilinear control for Gross-Pitaevskii equations with singular potentials

Kai Wang, Dun Zhao

Abstract:
We study the optimal bilinear control problem of the generalized Gross-Pitaevskii equation

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.

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Kai Wang
School of Mathematics and Statistics
Lanzhou University
Lanzhou 730000, China
email: wkai558@163.com
Dun Zhao
School of Mathematics and Statistics
Lanzhou University
Lanzhou 730000, China
email: zhaod@lzu.edu.cn

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