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Properties of Measure Controls and Their Trajectories

Mauro Garavello, Xiaoqian Gong, Benedetto Piccoli

TL;DR

This paper develops a transport-based framework for control of measures by introducing measure controls and measure vector fields within measure differential equations ($MDE$). It proves existence and well-posedness of systems with measure controls and establishes a one-to-one correspondence between measure controls and measure vector fields, also showing stability and closure of the trajectory set in the Wasserstein topology. The approach relies on Lattice Approximation Solutions (LAS), measure disintegration, and selection theorems to build $V^{\tilde{u}}$ and connect MVFs with measure controls. The results provide a rigorous foundation for uncertainty-robust control of distributed systems and have potential applications to crowd dynamics, population biology, and multi-agent coordination.

Abstract

This paper deals with the concepts of measure controls and of measure vector fields, within the mathematical framework of measure differential equations (MDEs), recently proposed in~\cite{piccoli_measure_2019}. Measure controls can be seen as a generalization of relaxed control. Moreover, they are particularly suitable for studying dynamics with uncertainty. The main results of this paper include establishing the existence and well-posedness of control systems with measure controls and proving the equivalence between measure controls and measure vector fields. The stability and closure properties of the trajectory set are also studied.

Properties of Measure Controls and Their Trajectories

TL;DR

This paper develops a transport-based framework for control of measures by introducing measure controls and measure vector fields within measure differential equations (). It proves existence and well-posedness of systems with measure controls and establishes a one-to-one correspondence between measure controls and measure vector fields, also showing stability and closure of the trajectory set in the Wasserstein topology. The approach relies on Lattice Approximation Solutions (LAS), measure disintegration, and selection theorems to build and connect MVFs with measure controls. The results provide a rigorous foundation for uncertainty-robust control of distributed systems and have potential applications to crowd dynamics, population biology, and multi-agent coordination.

Abstract

This paper deals with the concepts of measure controls and of measure vector fields, within the mathematical framework of measure differential equations (MDEs), recently proposed in~\cite{piccoli_measure_2019}. Measure controls can be seen as a generalization of relaxed control. Moreover, they are particularly suitable for studying dynamics with uncertainty. The main results of this paper include establishing the existence and well-posedness of control systems with measure controls and proving the equivalence between measure controls and measure vector fields. The stability and closure properties of the trajectory set are also studied.
Paper Structure (12 sections, 12 theorems, 91 equations)

This paper contains 12 sections, 12 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Control Systems
  3. Differential Inclusions
  4. Relaxed Controls
  5. Random and Measure Vector Fields
  6. Measure Controls
  7. Equivalence of MVF and Measure Controls
  8. Peano-type Theorem for Measure Controls
  9. Well-Posedness Result for Measure Controls
  10. Stability of Trajectories
  11. Closure of Trajectories
  12. Probability Measures and Wasserstein Distance

Key Result

Proposition 1

Assume ass:compact_controls and ass:control_system. Then, the map $x \mapsto F(x)$, defined in eq:diff_inclusion, is Lipschitz continuous for the Hausdorff metric $d_H$, that is for every $x, y \in \mathbb{R}^n$, where $L_f$ is the Lipschitz constant of $f$.

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Definition 4
  • Proposition 3
  • Definition 5
  • Definition 6
  • ...and 14 more