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Sparse optimal control in the Wasserstein space

Enrico Sartor, Florian Dörfler, Nicolas Lanzetti

TL;DR

Under suitable assumptions on the system dynamics and Wasserstein differentiability of the terminal cost, this work proves first-order sensitivity of the control-to-state map, derive an adjoint system and an explicit formula for the gradient of the cost functional, and obtains Pontryagin-type necessary conditions.

Abstract

We study sparse optimal control of a non-local continuity equation, where the goal is to steer a distribution via finitely many controllable agents or actuators. This model arises naturally in mean-field multi-agent systems and takes the form of a coupled PDE-ODE system where the PDE describes the evolution of the distribution and the controlled ODE captures the dynamics of the controllable agents. A natural objective is distribution steering via terminal costs based on optimal transport, such as the squared Wasserstein distance. These costs are problematic for finite-agent formulations due to non-smoothness at empirical measures and they fall outside common expected-value-type cost classes. We address these challenges by studying the resulting optimal control problem in the Wasserstein space. Under suitable assumptions on the system dynamics and Wasserstein differentiability of the terminal cost (with no smoothness requirement on the associated Wasserstein gradient), we prove first-order sensitivity of the control-to-state map, derive an adjoint system and an explicit formula for the gradient of the cost functional, and obtain Pontryagin-type necessary conditions. To illustrate the resulting adjoint-based method, we present numerical experiments on a representative distribution-splitting task.

Sparse optimal control in the Wasserstein space

TL;DR

Under suitable assumptions on the system dynamics and Wasserstein differentiability of the terminal cost, this work proves first-order sensitivity of the control-to-state map, derive an adjoint system and an explicit formula for the gradient of the cost functional, and obtains Pontryagin-type necessary conditions.

Abstract

We study sparse optimal control of a non-local continuity equation, where the goal is to steer a distribution via finitely many controllable agents or actuators. This model arises naturally in mean-field multi-agent systems and takes the form of a coupled PDE-ODE system where the PDE describes the evolution of the distribution and the controlled ODE captures the dynamics of the controllable agents. A natural objective is distribution steering via terminal costs based on optimal transport, such as the squared Wasserstein distance. These costs are problematic for finite-agent formulations due to non-smoothness at empirical measures and they fall outside common expected-value-type cost classes. We address these challenges by studying the resulting optimal control problem in the Wasserstein space. Under suitable assumptions on the system dynamics and Wasserstein differentiability of the terminal cost (with no smoothness requirement on the associated Wasserstein gradient), we prove first-order sensitivity of the control-to-state map, derive an adjoint system and an explicit formula for the gradient of the cost functional, and obtain Pontryagin-type necessary conditions. To illustrate the resulting adjoint-based method, we present numerical experiments on a representative distribution-splitting task.
Paper Structure (33 sections, 12 theorems, 83 equations, 4 figures, 1 algorithm)

This paper contains 33 sections, 12 theorems, 83 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Related work
  4. Notation
  5. Background material
  6. Wasserstein distance
  7. Differentiable structure of $\mathcal{P}_2(\mathbb{R}^n)$
  8. The controlled dynamics: well-posedness and continuous dependence
  9. Well-posedness of the controlled system
  10. Continuous dependence
  11. Differentiation of the dynamics
  12. The linearized dynamics
  13. Derivative of the flow
  14. Proof of \ref{['thrm: differentiability']}
  15. The optimal control problem
  16. ...and 18 more sections

Key Result

Proposition 1.2

If $\varphi\colon \mathcal{P}_2(\mathbb{R}^n)\to\mathbb{R}$ is Wasserstein differentiable at $\mu$ then for every $\nu$ we have that for every $\gamma\in\Gamma(\mu,\nu)$.

Figures (4)

  • Figure 1: As an example of our problem setting, consider two controlled agents (red circles) that "split" a distribution into two of equal mass located at two target locations.
  • Figure 2: Optimal trajectories of the controlled agents (red circles) that split the distribution (in blue) into two over the desired targets (black diamonds).
  • Figure 3: Trajectories in the case of finitely many agents. The behavior closely mimics the one in the mean-field, with successful "splitting" of the distribution.
  • Figure :

Theorems & Definitions (25)

  • Definition 1.1: Wasserstein differentiability
  • Proposition 1.2
  • Proposition 1.3
  • Definition 2.1: Weak measure solution
  • Theorem 2.2: Well-posedness and flow representation
  • Proposition 2.3
  • Theorem 2.4: Continuous dependence
  • proof
  • Remark 3.1
  • Proposition 3.2
  • ...and 15 more