  Bose-Einstein condensates (BECs) offer the potential to examine quantum
behavior at large length and time scales, as well as forming promising
candidates for quantum technology applications. Thus, the manipulation of BECs
using control fields is a topic of prime interest. We consider BECs in the mean
field model of the Gross-Pitaevskii equation (GPE), which contains linear and
nonlinear features, both of which are subject to control. In this work we
report successful optimal control simulations of a one dimensional GPE by
modulating the linear and nonlinear terms to stimulate transitions into excited
coherent modes. The linear and nonlinear controls are allowed to freely vary
over space and time to seek their optimal forms. The determination of the
excited coherent modes targeted for optimization is numerically performed
through an adaptive imaginary time propagation method. Numerical simulations
are performed for optimal control of mode-to-mode transitions between the
ground coherent mode and excited modes of a BEC trapped in a harmonic well. The
results show greater than 99% success for nearly all trials utilizing
reasonable initial guesses for the controls, and analysis of the optimal
controls reveals primarily direct transitions between initial and target modes.
The success of using solely the nonlinearity term as a control opens up further
research toward exploring novel control mechanisms inaccessible to linear
Schr\"odinger-type systems.
