[Paper Review] Differential Inclusions in Wasserstein Spaces: The Cauchy-Lipschitz Framework

Benoît Bonnet, Hélène Frankowska|arXiv (Cornell University)|Jul 17, 2020

Geometric Analysis and Curvature Flows68 references41 citations

TL;DR

The paper develops a functional framework for differential inclusions in Wasserstein spaces, defining solutions via measurable selections of velocity fields and proving Filippov, Relaxation, and compactness results to study mean-field control problems.

ABSTRACT

In this article, we propose a general framework for the study of differential inclusions in the Wasserstein space of probability measures. Based on earlier geometric insights on the structure of continuity equations, we define solutions of differential inclusions as absolutely continuous curves whose driving velocity fields are measurable selections of multifunction taking their values in the space of vector fields. In this general setting, we prove three of the founding results of the theory of differential inclusions: Filippov's theorem, the Relaxation theorem, and the compactness of the solution sets. These contributions -- which are based on novel estimates on solutions of continuity equations -- are then applied to derive a new existence result for fully non-linear mean-field optimal control problems with closed-loop controls.

Motivation & Objective

• Motivate the study of large dynamical systems and mean-field limits in Wasserstein spaces.
• Define differential inclusions in Wasserstein spaces via measurable selections of velocity fields.
• Establish foundational results analogous to Filippov’s theorem, the Relaxation theorem, and compactness of solution sets.
• Apply the framework to obtain existence results for fully nonlinear mean-field optimal control with closed-loop controls.

Proposed method

• Introduce set-valued velocity fields V on [0,T] and define differential inclusions as ∂tμ(t) ∈ −div(V(t,μ(t))μ(t)).
• Interpret the velocity as a projection of nonlocal velocities onto the Wasserstein tangent space.
• Prove generalisations of Filippov’s theorem, the Relaxation theorem, and compactness of trajectory sets in (P_c(ℝ^d),Wp).
• Derive a priori estimates for solutions of continuity equations to support stability results.
• Use measurable selection theorems to relate selections in V to actual control functions u(t) that drive characteristics.
• Apply results to obtain existence for constrained mean-field optimal control problems.

Experimental results

Research questions

• RQ1 How to formulate differential inclusions in Wasserstein spaces using a set-valued velocity field?
• RQ2 Can Filippov’s theorem be extended to Cauchy-Lipschitz differential inclusions in Wasserstein spaces?
• RQ3 Does a Relaxation theorem hold for differential inclusions in Wasserstein spaces and what does its closure look like?
• RQ4 Are solution sets to these inclusions compact under convexity assumptions on the right-hand side?
• RQ5 How can these theoretical results yield existence results for fully nonlinear mean-field optimal control with closed-loop controls?

Key findings

• A robust notion of solutions to differential inclusions in Wasserstein spaces is established via measurable selections of locally Lipschitz velocity fields.
• Filippov-type results are extended to (P_c(ℝ^d), Wp), yielding Grönwall-type inequalities for trajectory selections.
• A Relaxation theorem is proven, describing the closure of the solution set when the right-hand side is convexified.
• Compactness of the solution set is shown under convex right-hand sides, enabling existence results for constrained mean-field control.
• New momentum and Grönwall-type estimates for continuity equations underpin the stability and convergence analyses.