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 language | English (US) |
|---|---|
| Article number | 123437 |
| Journal | Journal of Mathematical Analysis and Applications |
| Volume | 481 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 1 2020 |
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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 journal › Article
TY - JOUR
T1 - Generalized dynamic programming principle and sparse mean-field control problems
AU - Cavagnari, Giulia
AU - Marigonda, Antonio
AU - Piccoli, Benedetto
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
UR - http://www.scopus.com/inward/record.url?scp=85071568262&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85071568262&partnerID=8YFLogxK
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 -