  We address the problem of 2D particle density control. The particles are
immersed in dielectric fluid and acted upon by manipulating an electric field.
The electric field is controlled by an array of electrodes and used to bring
the particle density to a desired pattern using dielectrophoretic forces. We
use a lumped, 2D, capacitive-based, nonlinear model describing the motion of a
particle. The spatial dependency of the capacitances is estimated using
electrostatic COMSOL simulations. We formulate an optimal control problem,
where the loss function is defined in terms of the error between the particle
density at some final time and a target density. We use a kernel density
estimator (KDE) as a proxy for the true particle density. The KDE is computed
using the particle positions that are changed by varying the electrode
potentials. We showcase our approach through numerical simulations, where we
demonstrate how the particle positions and the electrode potentials vary when
shaping the particle positions from a uniform to a Gaussian distribution.
