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

2026-03-19

AI Summary

This work addresses the high computational complexity inherent in achieving efficient spatial density distributions in multi-agent systems, where conventional density-driven optimal control strategies are often intractable for online implementation. To overcome this challenge, the authors propose a dimensionality reduction approach based on analytical Karush–Kuhn–Tucker (KKT) conditions, which reformulates the multi-step predictive control problem into a quadratic program with linear time complexity O(T). This method is integrated within a contractive model predictive control (MPC) framework, incorporating Lyapunov-based contraction constraints to guarantee input-to-state stability of the closed-loop system. The resulting algorithm substantially reduces computational overhead, enabling real-time density regulation for large-scale multi-agent systems over extended prediction horizons. Numerical simulations demonstrate its superior performance in rapid spatial coverage and computational efficiency.

Technology Category

• Multiagent Systems: Multiagent Planning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• multi-agent systems
• density-driven control
• computational complexity
• optimal control
• spatial distribution

Innovation

• Density-Driven Optimal Control
• Analytical KKT Reduction
• Contractive MPC
• Input-to-State Stability
• Computational Scalability