  In this paper, we newly formulate and solve the optimal density control
problem with Gromov-Wasserstein (GW) terminal cost in discrete-time linear
Gaussian systems. Differently from the Wasserstein or Kullback-Leibler
distances employed in the existing works, the GW distance quantifies the
difference in shapes of the distribution, which is invariant under translation
and rotation. Consequently, our formulation allows us to find small energy
inputs that achieve the desired shape of the terminal distribution, which has
practical applications, e.g., robotic swarms. We demonstrate that the problem
can be reduced to a Difference of Convex (DC) programming, which is efficiently
solvable through the DC algorithm. Through numerical experiments, we confirm
that the state distribution reaches the terminal distribution that can be
realized with the minimum control energy among those having the specified
shape.
