Generalized dynamic programming principle and sparse mean-field control problems

Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Article number123437
JournalJournal of Mathematical Analysis and Applications
Volume481
Issue number1
DOIs
StatePublished - Jan 1 2020

Fingerprint

Dynamic Programming Principle
Dynamic programming
Mean Field
Control Problem
Optimal Trajectory
Continuity Equation
Particle System
Hamilton-Jacobi Equation
Viscosity Solutions
Sparsity
Value Function
Velocity Field
Instantaneous
Cost Function
Optimal Control Problem
Cost functions
Restriction
Minimise
Norm
Trajectories

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Keywords

  • Control with uncertainty
  • Dynamic programming principle
  • Hamilton-Jacobi equation in Wasserstein space
  • Multi-agent mean field sparse control

Cite this

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Generalized dynamic programming principle and sparse mean-field control problems. / Cavagnari, Giulia; Marigonda, Antonio; Piccoli, Benedetto.

In: Journal of Mathematical Analysis and Applications, Vol. 481, No. 1, 123437, 01.01.2020.

Research output: Contribution to journalArticle

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AU - Marigonda, Antonio

AU - Piccoli, Benedetto

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