Optimal bilinear control of Gross–Pitaevskii equations with Coulombian potentials

In this paper, we consider an optimal bilinear control problem for the Gross–Pitaevskii equations with Coulombian potentials. We show the well-posedness of the problem and the existence of an optimal control. In addition, the first order optimality system is rigorously derived. In particular, we prove the Fréchet-differentiability of the unconstrained functional. We extend the study of Hintermüller et al. (2013) [15] to more general power nonlinearities and unbounded potentials.