Optimal bilinear control for Gross-Pitaevskii equations with singular potentials

Date

2019-10-13

Authors

Wang, Kai
Zhao, Dun

Journal Title

Journal ISSN

Volume Title

Publisher

Texas State University, Department of Mathematics

Abstract

We study the optimal bilinear control problem of the generalized Gross-Pitaevskii equation i∂tu = -∆u + U(x)u + φ(t) 1/|x|α u + λ|u|2σu, x ∈ ℝ3, where U(x) is the given external potential, φ(t) is the control function. The existence of an optimal control and the optimality condition are presented for suitable α and σ. In particular, when 1 ≤ α < 3/2, the Fréchet-differentiability of the objective functional is proved for two cases: (i) λ < 0, 0 < σ < 2/3; (ii) λ > 0, 0 < σ < 2. Comparing with the previous studies in [6], the results fill the gap for σ ∈ (0, 1/2).

Description

Keywords

Optimal bilinear control, Gross-Pitaevskii equation, Objective functional, Frechet-differentiability, Optimal condition

Citation

Wang, K., & Zhao, D. (2019). Optimal bilinear control for Gross-Pitaevskii equations with singular potentials. <i>Electronic Journal of Differential Equations, 2019</i>(115), pp. 1-13.

Rights

Attribution 4.0 International

Rights Holder

Rights License