Computationally Efficient Density-Driven Optimal Control via Analytical KKT Reduction and Contractive MPC

TL;DR

This paper introduces a computationally efficient method for density-driven optimal control in multi-agent systems by reducing KKT system complexity and ensuring stability, enabling scalable long-horizon predictive control.

Contribution

It presents an analytical KKT reduction transforming the large-scale system into a linear-scaling quadratic program, with stability guarantees for dynamic environments.

Findings

Achieves O(T) computational complexity for predictive control

Ensures closed-loop stability via Lyapunov constraints

Enables rapid density coverage in large-scale multi-agent swarms

Abstract

Efficient coordination for collective spatial distribution is a fundamental challenge in multi-agent systems. Prior research on Density-Driven Optimal Control (D2OC) established a framework to match agent trajectories to a desired spatial distribution. However, implementing this as a predictive controller requires solving a large-scale Karush-Kuhn-Tucker (KKT) system, whose computational complexity grows cubically with the prediction horizon. To resolve this, we propose an analytical structural reduction that transforms the T-horizon KKT system into a condensed quadratic program (QP). This formulation achieves O(T) linear scalability, significantly reducing the online computational burden compared to conventional O(T^3) approaches. Furthermore, to ensure rigorous convergence in dynamic environments, we incorporate a contractive Lyapunov constraint and prove the Input-to-State Stability (ISS) of the closed-loop system against reference propagation drift. Numerical simulations verify that the proposed method facilitates rapid density coverage with substantial computational speed-up, enabling long-horizon predictive control for large-scale multi-agent swarms.