Optimal bilinear control for Gross-Pitaevskii equations with singular potentials

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).