TY - JOUR
T1 - Generalized dynamic programming principle and sparse mean-field control problems
AU - Cavagnari, Giulia
AU - Marigonda, Antonio
AU - Piccoli, Benedetto
N1 - Funding Information:
The authors acknowledge the partial support of the NSF Project Kinetic description of emerging challenges in multiscale problems of natural sciences, DMS Grant # 1107444 and the endowment fund of the Joseph and Loretta Lopez Chair . B.P. and G.C. thank NSF for support via the CPS Synergy projects CNS 1446715 and CNS 1837481 .
Funding Information:
This work has been partially supported by the project of the Italian Ministry of Education, Universities and Research (MIUR) “Dipartimenti di Eccellenza 2018-2022”.
Funding Information:
G.C. has been supported by Cariplo Foundation and Regione Lombardia through the project 2016-2018 “Variational evolution problems and optimal transport”. G.C. thanks the Department of Mathematical Sciences of Rutgers University - Camden (U.S.A.).
PY - 2020/1/1
Y1 - 2020/1/1
N2 - 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.
AB - 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.
KW - Control with uncertainty
KW - Dynamic programming principle
KW - Hamilton-Jacobi equation in Wasserstein space
KW - Multi-agent mean field sparse control
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U2 - 10.1016/j.jmaa.2019.123437
DO - 10.1016/j.jmaa.2019.123437
M3 - Article
AN - SCOPUS:85071568262
VL - 481
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
M1 - 123437
ER -