Minimum energy density steering of linear systems with Gromov-Wasserstein terminal cost

In this study, we address optimal control problems focused on steering the probabilistic distribution of state variables in linear dynamical systems. Specifically, we address the problem of controlling the structural properties of Gaussian state distributions to predefined targets at terminal times. This task is not yet explored in existing works that primarily aim to exactly match state distributions. By employing the Gromov-Wasserstein (GW) distance as the terminal cost, we formulate a problem that seeks to align the structure of the state density with that of a desired distribution. This approach allows us to extend the control objectives to capture the distribution's shape. We demonstrate that this complex 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 terminal distribution indeed gets closer to the desired structural properties of the target distribution.