The Ground Cost for Optimal Transport of Angular Velocity
This work addresses the problem of controlling spin states in rigid bodies like spacecraft under stochastic conditions, but it is incremental as it builds on prior existence-uniqueness results by focusing on structural aspects of the formulation.
The authors derived the ground cost for the Kantorovich formulation of an optimal transport problem over angular velocity dynamics, enabling stochastic guidance of rigid body spin states under deadline constraints. They showed this reduces to solving a structured deterministic nonlinear optimal control problem, applicable to a class of generalized optimal mass transport problems with nonlinear dynamics.
We revisit the optimal transport problem over angular velocity dynamics given by the controlled Euler equation. The solution of this problem enables stochastic guidance of spin states of a rigid body (e.g., spacecraft) over a hard deadline constraint by transferring a given initial state statistics to a desired terminal state statistics. This is an instance of generalized optimal transport over a nonlinear dynamical system. While prior work has reported existence-uniqueness and numerical solution of this dynamical optimal transport problem, here we present structural results about the equivalent Kantorovich a.k.a. optimal coupling formulation. Specifically, we focus on deriving the ground cost for the associated Kantorovich optimal coupling formulation. The ground cost is equal to the cost of transporting unit amount of mass from a specific realization of the initial or source joint probability measure to a realization of the terminal or target joint probability measure, and determines the Kantorovich formulation. Finding the ground cost leads to solving a structured deterministic nonlinear optimal control problem, which is shown to be amenable to an analysis technique pioneered by Athans et al. We show that such techniques have broader applicability in determining the ground cost (thus Kantorovich formulation) for a class of generalized optimal mass transport problems involving nonlinear dynamics with translated norm-invariant drift.