Generalized dynamic programming principle and sparse mean-field control problems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:481

Article

Cavagnari, Giulia¹  Marigonda, Antonio²  Piccoli, Benedetto³ 

[1] Univ Pavia, Dept Math F Casorati, Via Ferrate 5, I-27100 Pavia, Italy

[2] Univ Verona, Dept Comp Sci, Str Le Grazie 15, I-37134 Verona, Italy

[3] Rutgers Univ Camden, Dept Math Sci, 311 N 5th St, Camden, NJ 08102 USA

关键词: Multi-agent mean field sparse control;    Hamilton-Jacobi equation in;    Wasserstein space;    Control with uncertainty;    Dynamic programming principle;   

DOI  :  10.1016/j.jmaa.2019.123437

来源: Elsevier

【 摘 要 】

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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free