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

Julian Martinez, Kooktae Lee

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.

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

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.
Paper Structure (19 sections, 5 theorems, 27 equations, 2 figures)

This paper contains 19 sections, 5 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Multi-Agent System Modeling
  4. Decentralized Density-Driven Optimal Control (D$^2$OC)
  5. Computational Efficiency Improvement via Multi-Step Reference Structural Reduction
  6. From Fixed Target to Multi-Stage Prediction
  7. Analytical Condensation to $m$-Dimensional Quadratic Form
  8. Computational Efficiency Analysis
  9. Contractive Lyapunov Constraint for Receding-Horizon D$^2$OC
  10. LMI-Based Stability Design via Schur Complement
  11. Numerical Reformulation for Real-Time Execution
  12. QCQP Formulation
  13. Closed-Loop Stability Analysis
  14. Numerical Resolution via Soft-Constrained QCQP
  15. Dual-Newton Solver for Real-Time Execution
  16. ...and 4 more sections

Key Result

Theorem 1

For the LTI system eqn: LTI system, the optimal control sequence $\bar{u}_i^k:=[(u_i^k)^{\top},\cdots,(u_i^{k+T-1})^{\top}]^{\top}$ minimizing eqn: minimization problem is analytically given by: where $\mathcal{E}_{ij}$ denotes the $(i,j)$-th block submatrix of $E^{-1}$ and the constituent blocks of the KKT matrix $E$ are defined as: Moreover, $F_1$ and $F_2$ in eqn: optimal_u_2 are defined by:

Figures (2)

  • Figure 1: Multi-agent coverage trajectories via D$^2$OC: (a) Full KKT baseline, (b) proposed reduced KKT formulation showing identical optimality, and (c) reduced KKT with recursive stability constraints.
  • Figure 2: Computation time comparison b/w the full KKT and the reduced KKT with stability guarantee.

Theorems & Definitions (12)

  • Theorem 1: Analytical Solution seo2025smcs
  • Remark 1
  • Definition 1: Iterative Sample Set and Weight Propagation
  • Theorem 2: Analytical Reduction of Time-Varying MPC
  • proof
  • Proposition 1: Computational Efficiency of Reduced KKT
  • proof
  • Proposition 2: Schur-Complement Contractive Constraint
  • proof
  • Theorem 3: Input-to-State Stability of Condensed D$^2$OC
  • ...and 2 more