Optimal bilinear control for Gross-Pitaevskii equations with singular potentials

Abstract

We study the optimal bilinear control problem of the generalized Gross-Pitaevskii equation $$ i\partial_{t}u=-\Delta u+U(x)u+\phi(t)\frac{1}{|x|^{\alpha}}u +\lambda|u|^{2\sigma}u,\quad x\in \mathbb{R}^3, $$ 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 $1\leq\alpha0$, $0<\sigma<2$. Comparing with the previous studies in [6], the results fill the gap for $\sigma \in (0,1/2)$.

Keywords

• optimal bilinear control
• gross-pitaevskii equation
• objective functional
• frechet-differentiability
• optimal condition