### 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 |

### Fingerprint

### 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

*Journal of Mathematical Analysis and Applications*,

*481*(1), [123437]. https://doi.org/10.1016/j.jmaa.2019.123437

}

*Journal of Mathematical Analysis and Applications*, vol. 481, no. 1, 123437. https://doi.org/10.1016/j.jmaa.2019.123437

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

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 -