New Control Laguerre Tessellation Optimizes Multi-Agent Transport with Optimal Control Costs

In a new arXiv paper, Ripon C. Sarker and Abhishek Halder tackle a frontier problem: optimally transporting a continuous distribution of controlled agents to discrete target locations, where the cost of moving each agent is determined by its optimal control problem (e.g., minimum energy or minimum time). They prove that when the ground cost satisfies the twist condition, the optimal transport map is given almost everywhere by a Laguerre tessellation of the state space—a geometric partition that assigns each region to the nearest target under a weighted distance metric. They call this structure a Control Laguerre Tessellation (CLT).

The authors illustrate CLT using two classic linear control objectives: minimum-energy and minimum-time. For minimum-energy, the ground cost becomes a quadratic function of state and control, and the tessellation adapts to the control effort required. For minimum-time, the cost is non-smooth, yet the twist condition holds, yielding a unique partition. This work bridges optimal transport and control theory, offering a principled way to design decentralized multi-agent systems, such as drone swarms recharging at stations or robots assembling components. The framework could also improve resource allocation in logistics or data center load balancing where agents must move under dynamic constraints.