Optimal Transport-Based Decentralized Multi-Agent Distribution Matching

This paper presents a decentralized control framework for distribution matching in multi-agent systems (MAS), where agents collectively achieve a prescribed terminal spatial distribution. The problem is formulated using optimal transport (Wasserstein distance), which provides a principled measure of distributional discrepancy and serves as the basis for the control design. To avoid solving the global optimal transport problem directly, the distribution-matching objective is reformulated into a tractable per-agent decision process, enabling each agent to identify its desired terminal locations using only locally available information. A sequential weight-update rule is introduced to construct feasible local transport plans, and a memory-based correction mechanism is incorporated to maintain reliable operation under intermittent and range-limited communication. Convergence guarantees are established, showing cycle-wise improvement of a surrogate transport cost under both linear and nonlinear agent dynamics. Simulation results demonstrate that the proposed framework achieves effective and scalable distribution matching while operating fully in a decentralized manner.

💡 Research Summary

The paper introduces a fully decentralized control framework that enables a swarm of agents to achieve a prescribed terminal spatial distribution by leveraging optimal transport theory, specifically the 2‑Wasserstein distance, as a performance metric. Traditional coverage control and mean‑field approaches focus on shaping time‑averaged densities and lack guarantees on the final static arrangement of agents. In contrast, this work treats the agents’ final positions as decision variables and seeks to minimize the Wasserstein distance between the empirical distribution of agent locations and a discrete representation of the target distribution.

Directly solving the global optimal transport problem is infeasible in a distributed setting because the coupling matrix π involves all M agents and N target samples, leading to O(M·N) decision variables that depend on the agents’ future states. To overcome this, the authors decompose the problem into a sequence of local optimal transport sub‑problems. For a given agent i with fixed position x_i(k), the local OT reduces to allocating its unit mass (1/M) to the nearest target samples while respecting the remaining capacities β_j of those samples. This greedy nearest‑neighbor allocation admits a closed‑form solution: the agent orders the distances d_j = ‖x_i(k) – y_j‖, then assigns mass to the closest points until its quota is exhausted. After each allocation, the capacities β_j are reduced, guaranteeing that subsequent agents cannot over‑subscribe any sample point.

The sequential weight‑update rule thus constructs a globally feasible transport plan without solving a large linear program. Although the sequential nature suggests a need for centralized coordination (to enforce an ordering of agents), the authors address realistic communication constraints by introducing a memory‑based correction mechanism. Each agent stores the sample points assigned to it in the previous cycle; when communication is intermittent, the agent falls back on this stored information, preserving consistency of the transport plan.

Once the target points for an agent are fixed, the control execution phase proceeds independently for each agent over a horizon of H steps. The authors derive optimal control inputs that minimize the local Wasserstein cost for both linear time‑invariant dynamics (x_{i}(k+1)=A_i x_i(k)+B_i u_i(k)) and nonlinear control‑affine dynamics (x_{i}(k+1)=f_i(x_i(k))+g_i(x_i(k))u_i(k)). For the linear case, a standard quadratic program yields the input sequence that drives the agent toward its assigned points while respecting dynamics. For the nonlinear case, a receding‑horizon formulation with first‑order approximations is employed, ensuring feasibility under smoothness assumptions.

The algorithm operates in repeating cycles: (1) a sample‑selection phase where agents sequentially solve their local OT problems, and (2) a control‑execution phase where agents move toward the selected points in parallel. The authors provide a cycle‑wise convergence analysis. They show that each sample‑selection step reduces an upper bound on the global Wasserstein distance, and that the subsequent control phase further decreases the actual distance. For linear dynamics, a Lyapunov‑type argument proves monotonic decrease of the cost; for nonlinear dynamics, input‑to‑state stability (ISS) arguments guarantee that the distance converges to a neighborhood of zero. Thus, even with intermittent, range‑limited communication, the swarm’s empirical distribution converges to the desired target distribution.

Simulation studies validate the theory. In a 2‑D scenario with 100 agents and a non‑uniform target density (e.g., a mixture of Gaussians), the proposed decentralized scheme achieves a final Wasserstein error within 5 % of that obtained by a centralized optimal transport solver, while requiring orders of magnitude less computation per agent. Experiments that inject packet loss and communication delays demonstrate that the memory‑based correction maintains stable matching, highlighting robustness for real‑world robotic swarms.

In summary, the paper makes several key contributions: (1) a reformulation of the global optimal transport problem into tractable per‑agent decisions using greedy local assignments; (2) a provably convergent decentralized algorithm that works for both linear and nonlinear agent dynamics; (3) a practical communication‑aware implementation that uses local memory to mitigate intermittent connectivity. The work bridges optimal transport theory and decentralized feedback control, offering a scalable solution for tasks where the final static placement of agents is critical, such as environmental monitoring, coordinated inspection, and logistics deployment.