Generalized Dynamic Programming Principle and Sparse Mean-Field Control Problems

TL;DR

This paper develops a generalized dynamic programming framework for sparse control problems in Wasserstein spaces, addressing both instantaneous and cumulative control constraints, with proofs of existence and viscosity solutions for the associated value functions.

Contribution

It introduces a unified approach to sparse mean-field control problems with new existence results and a dynamic programming principle in Wasserstein spaces.

Findings

Proves existence of optimal trajectories under control sparsity constraints.

Establishes the value function as a viscosity solution of a Hamilton-Jacobi-Bellman equation.

Provides an abstract dynamic programming principle applicable to these control problems.

Abstract

In this paper we study optimal control problems in Wasserstein spaces, which are suitable to describe macroscopic dynamics of multi-particle systems. The dynamics is described by a parametrized continuity equation, in which the Eulerian velocity field is affine w.r.t. some variables. Our aim is to minimize a cost functional which includes a control norm, thus enforcing a \emph{control sparsity} constraint. More precisely, we consider a nonlocal restriction on the total amount of control that can be used depending on the overall state of the evolving mass. We treat in details two main cases: an instantaneous constraint on the control applied to the evolving mass and a cumulative constraint, which depends also on the amount of control used in previous times. For both constraints, we prove the existence of optimal trajectories for general cost functions and that the value function is viscosity solution of a suitable Hamilton-Jacobi-Bellmann equation. Finally, we discuss an abstract Dynamic Programming Principle, providing further applications in the Appendix.