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Optimal control problems driven by nonlinear degenerate Fokker-Planck equations

Francesca Anceschi, Giacomo Ascione, Daniele Castorina, Francesco Solombrino

Abstract

The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as mean-field limits of Stochastic Differential models for multipopulation dynamics, where a large number of agents (followers) is steered through parsimonious intervention on a selected class of leaders. The proposed approach combines stability estimates for measure solutions of nonlinear degenerate Fokker-Planck equations with a general framework of assumptions on the cost functional, ensuring compactness and lower semicontinuity properties. The Lie structure of the state equations allows one for considering non-Lipschitz nonlinearities, provided some suitable dissipativity assumptions are considered in addition to non-Euclidean Hölder and sublinearity conditions.

Optimal control problems driven by nonlinear degenerate Fokker-Planck equations

Abstract

The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as mean-field limits of Stochastic Differential models for multipopulation dynamics, where a large number of agents (followers) is steered through parsimonious intervention on a selected class of leaders. The proposed approach combines stability estimates for measure solutions of nonlinear degenerate Fokker-Planck equations with a general framework of assumptions on the cost functional, ensuring compactness and lower semicontinuity properties. The Lie structure of the state equations allows one for considering non-Lipschitz nonlinearities, provided some suitable dissipativity assumptions are considered in addition to non-Euclidean Hölder and sublinearity conditions.

Paper Structure

This paper contains 31 sections, 20 theorems, 325 equations.

Table of Contents

  1. Introduction
  2. Presentation of the problem
  3. Our assumptions
  4. Main results
  5. Structure of our work
  6. Useful notation and preliminary results
  7. Wasserstein spaces
  8. Spaces of continuous functions and related Wasserstein spaces
  9. Random variables and couplings
  10. Young functions
  11. The linear equation
  12. Proof of Theorem \ref{['thm:main1']} and Corollary \ref{['cor:stab']}
  13. Proof of Theorem \ref{['thm:main1']}
  14. Existence with $\mathfrak{v}$ uniformly sublinear
  15. Existence for a general $\mathfrak{v}$
  16. ...and 16 more sections

Key Result

Theorem 1.3

Let $p \ge 1$, $\overline{\mu} \in \mathcal{W}_p({{\mathbb R}}^{2d})$ and suppose Assumptions ass:v hold. Then the non linear Cauchy problem probmunl2-intro admits a unique global solution $\bm{\mu} \in C^{\gamma_p}_{\rm loc}({{\mathbb R}}_0^+;\mathcal{W}_p({{\mathbb R}}^{2d}))$, where $\gamma_p=\fr where $K$ is the constant defined in $(\mathfrak{v}_1)$.

Theorems & Definitions (39)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Remark 3.1
  • ...and 29 more