Control Laguerre Tessellation: Semi-discrete Optimal Transport Over Control Systems

2026-07-10

AI Summary

This work addresses the optimal transport problem from a continuous source distribution to a discrete target measure, where the transport cost is induced by the optimal control cost of an agent’s motion—such as minimal energy or shortest time. For such control-induced costs satisfying the twist condition, the authors introduce the Control Laguerre Tessellation (CLT), a generalization of the classical Laguerre tessellation to the optimal control setting. By integrating semi-discrete optimal transport theory with geometric construction techniques, CLT yields an optimal transport map defined almost everywhere. The framework is effectively implemented and validated on two classes of linear controlled systems, offering a novel computational tool for control-driven optimal transport problems.

Abstract

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.

Problem

• optimal transport
• control systems
• Laguerre tessellation
• ground cost
• discrete target measure

Innovation

• Control Laguerre Tessellation
• semi-discrete optimal transport
• twist condition
• optimal control
• Laguerre tessellation