Control Laguerre Tessellation: Semi-discrete Optimal Transport Over Control Systems
2026-07-10 • Machine Learning
Machine LearningMultiagent Systems
AI summaryⓘ
The authors study how to best move agents, who follow specific control rules, from a smooth starting distribution to a set of target points. They use the cost of moving the agents optimally to define distances, then show that if these costs meet a technical condition called the twist condition, the best way to transport agents can be described using a special kind of geometric partition called a Control Laguerre Tessellation (CLT). This approach generalizes how we usually divide space into regions based on distance, tailored to the agents' control rules. The authors demonstrate this theory using agents with linear controls aiming to minimize either energy or time.
Optimal transportControlled agentsGround costTwist conditionLaguerre tessellationControl Laguerre TessellationCompactly supported measureAbsolutely continuous measureOptimal controlLinear control systems
Authors
Ripon C. Sarker, Abhishek Halder
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.