Optimal consensus control of the Cucker-Smale model

TL;DR

This paper develops numerical methods for optimal consensus control in multi-agent Cucker-Smale systems, including gradient descent for smooth penalties and heuristics for sparse controls, with mean-field approximations for large agent populations.

Contribution

It introduces a comprehensive numerical framework for optimal consensus control, combining gradient methods and heuristics, and extends analysis to large systems via mean-field models.

Findings

Gradient descent effectively approximates optimal controls with smooth penalties.

Heuristic methods successfully realize sparse control strategies.

Mean-field models approximate large-agent consensus control problems.

Abstract

We study the numerical realisation of optimal consensus control laws for agent-based models. For a nonlinear multi-agent system of Cucker-Smale type, consensus control is cast as a dynamic optimisation problem for which we derive first-order necessary optimality conditions. In the case of a smooth penalization fo the control energy, the optimality system is numerically approximated via a gradient-descent method. For sparsity promoting, non-smooth $\ell_1$-norm control penalizations, the optimal controllers are realised by means of heuristic methods. For an increasing number of agents, we discuss the approximation of the consensus control problem by following a mean-field modelling approach.