Control Laguerre Tessellation: Semi-discrete Optimal Transport Over Control Systems
This provides a theoretical framework for optimal transport in control systems, relevant to researchers in optimal transport and control theory, but is primarily a theoretical extension without empirical validation.
The paper extends optimal transport to control systems, proving that the optimal transport map is given by a Laguerre tessellation of the state space when the ground cost satisfies the twist condition, and illustrates this for linear controlled agents with minimum energy and minimum time objectives.
We study the optimal transport of optimally controlled agents from a compactly supported absolutely continuous source to a discrete target measure. The ground cost for the transport is induced by the optimal cost of the agents' motion. When this ground cost satisfies the twist condition, the optimal transport map is given almost everywhere in terms of a Laguerre tessellation of the state space. We refer to this control-theoretic generalization of Laguerre tessellation as Control Laguerre Tessellation (CLT), and illustrate it for two ground costs induced by linear controlled agents with minimum energy and minimum time objectives.