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Optimal control for multiagent systems with simultaneous aggregation

Mauro Bonafini, Giulia Cavagnari, Antonio Marigonda

TL;DR

The paper develops an optimal-control framework for multi-agent systems in the Wasserstein space that favors simultaneous aggregation via a time-local multiplicity. Using a lifted path-space formulation and the superposition principle, it defines a non-local energy $\mathscr E_{\psi,\phi}$ and proves lower semicontinuity and existence of minimizers, together with a Dynamic Programming Principle for the value function $V_T(t,\mu)$. A horizon-dependent aggregation effect is illustrated through a concrete example showing how the meeting point and stickiness depend on the time horizon. The results provide a rigorous foundation for optimal control in Wasserstein spaces with non-local, time-sensitive costs and have potential applications in coordinated aggregation and related domains.

Abstract

In this paper, we introduce an optimal control problem for multi-agent systems with non-local cost which favors simultaneous aggregation of particles. This is done introducing a time-dependent notion of multiplicity whose intrinsic dynamical nature differs from more established geometric-like definitions.

Optimal control for multiagent systems with simultaneous aggregation

TL;DR

The paper develops an optimal-control framework for multi-agent systems in the Wasserstein space that favors simultaneous aggregation via a time-local multiplicity. Using a lifted path-space formulation and the superposition principle, it defines a non-local energy and proves lower semicontinuity and existence of minimizers, together with a Dynamic Programming Principle for the value function . A horizon-dependent aggregation effect is illustrated through a concrete example showing how the meeting point and stickiness depend on the time horizon. The results provide a rigorous foundation for optimal control in Wasserstein spaces with non-local, time-sensitive costs and have potential applications in coordinated aggregation and related domains.

Abstract

In this paper, we introduce an optimal control problem for multi-agent systems with non-local cost which favors simultaneous aggregation of particles. This is done introducing a time-dependent notion of multiplicity whose intrinsic dynamical nature differs from more established geometric-like definitions.

Paper Structure

This paper contains 4 sections, 8 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and settings
  3. Existence of minimizers
  4. Dynamic Programming Principle

Key Result

Lemma 3.1

Let $\mathbb X$ be a separable metric space, $\{g_n\}_{n\in\mathbb N}$ be a sequence of Borel functions $g_n:\mathbb X\to [0,+\infty]$, and $\{\theta_n\}_{n\in\mathbb N}\subseteq \mathscr P(\mathbb X)$, $\theta\in\mathscr P(\mathbb X)$ be such that $\theta_n\rightharpoonup\theta$ as $n\to +\infty$. where $g(x):=\mathop{\mathrm{min\newline}}\limits\left\{\liminf_{n\to +\infty}g_n(x),\liminf_{\subs

Theorems & Definitions (20)

  • Definition 2.1: Multiagent dynamics
  • Definition 2.2: Multiplicity
  • Definition 2.3: Cost functional and variational problem
  • Example 2.4
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 10 more