Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Control of Nonlocal Balance Equations

Nikolay Pogodaev, Maxim Staritsyn

TL;DR

The paper studies an optimal control problem for a nonlocal balance law on the space of nonnegative measures with compact support. It develops a reduction to a conservative continuity equation on an extended probability space via a barycentric projection, and then recasts the dynamics as a Hilbert-space evolution, enabling a Pontryagin maximum principle for the original problem. Two equivalent PMP formulations connect to mean-field control theory and facilitate analysis and numerics. The framework also suggests avenues for numerical algorithms and extensions to sliding-mode controls and differential inclusions in Hilbert spaces.

Abstract

The paper presents an approach to studying optimal control problems in the space of nonnegative measures with dynamics given by a nonlocal balance law. This approach relies on transforming the balance law into a continuity equation in the space of probabilities, and subsequently into an ODE in a Hilbert space. The main result is a version of Pontryagin's maximum principle for the addressed problem, which encompasses all known formulations of this type in mean field control theory.

Optimal Control of Nonlocal Balance Equations

TL;DR

The paper studies an optimal control problem for a nonlocal balance law on the space of nonnegative measures with compact support. It develops a reduction to a conservative continuity equation on an extended probability space via a barycentric projection, and then recasts the dynamics as a Hilbert-space evolution, enabling a Pontryagin maximum principle for the original problem. Two equivalent PMP formulations connect to mean-field control theory and facilitate analysis and numerics. The framework also suggests avenues for numerical algorithms and extensions to sliding-mode controls and differential inclusions in Hilbert spaces.

Abstract

The paper presents an approach to studying optimal control problems in the space of nonnegative measures with dynamics given by a nonlocal balance law. This approach relies on transforming the balance law into a continuity equation in the space of probabilities, and subsequently into an ODE in a Hilbert space. The main result is a version of Pontryagin's maximum principle for the addressed problem, which encompasses all known formulations of this type in mean field control theory.

Paper Structure

This paper contains 15 sections, 9 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Example: Opinion Dynamics
  4. Goals. Main Ideas. Contribution and Novelty
  5. Notations and Conventions
  6. Reduction to a continuity equation
  7. Calculus on the Spaces of Measures
  8. Assumptions
  9. State Equation
  10. Barycentric Projection
  11. Properties of the Barycentric Projection
  12. Problem Transformation
  13. Pontryagin's Maximum Principle
  14. Reduction to an Evolution Equation in a Hilbert space
  15. Conclusion

Key Result

Lemma 1

Let $u_{(\cdot)}$ and $\mu_{(\cdot)}$ be defined as above. Introduce a non-autonomous vector field $f\colon I \times \mathbb{R}^n \to \mathbb{R}^n$ and a function $g\colon I \times \mathbb{R}^n \to \mathbb{R}$ by: The action of measures $\mu_t$ on test functions $\varphi \in \mathcal{C}^1(\mathbb{R}^n)$ can be expressed as follows: where $t \mapsto X_{t}(x)$ is a unique solution to the ordinary

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Remark 1
  • Example 1
  • Lemma 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • ...and 7 more