On a mean-field Pontryagin minimum principle for stochastic optimal control
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910199684530176 |
|---|---|
| author | Opper, Manfred Reich, Sebastian |
| author_facet | Opper, Manfred Reich, Sebastian |
| contents | This paper outlines a novel extension of the classical Pontryagin minimum (maximum) principle to stochastic optimal control problems. Contrary to the well-known stochastic Pontryagin minimum principle involving forward-backward stochastic differential equations, the proposed formulation is deterministic and of mean-field type. We denote it by the McKean-Pontryagin minimum principle. The Hamiltonian structure of the proposed McKean-Pontryagin minimum principle is achieved via the introduction of a pair of auxiliary functions. A gauge freedom in the choice of one of these two functions can be used to decouple the forward and reverse time equations; hence simplifying the solution of the underlying boundary value problem. We also consider infinite horizon discounted cost optimal control problems. In this case, the mean-field formulation allows one to convert the computation of the desired optimal control law into solving a pair of forward mean-field ordinary differential equations. The McKean-Pontryagin minimum principle is tested numerically for a controlled inverted pendulum, a controlled Lorenz-63 system, and a controlled Lorenz-96 system. Although the focus is on linear-quadratic control problems, the proposed methodology is extendable to more general problems including mean-field type control formulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a mean-field Pontryagin minimum principle for stochastic optimal control Opper, Manfred Reich, Sebastian Optimization and Control Numerical Analysis 35F21, 49M99, 93E20, 70H30, 70H45 This paper outlines a novel extension of the classical Pontryagin minimum (maximum) principle to stochastic optimal control problems. Contrary to the well-known stochastic Pontryagin minimum principle involving forward-backward stochastic differential equations, the proposed formulation is deterministic and of mean-field type. We denote it by the McKean-Pontryagin minimum principle. The Hamiltonian structure of the proposed McKean-Pontryagin minimum principle is achieved via the introduction of a pair of auxiliary functions. A gauge freedom in the choice of one of these two functions can be used to decouple the forward and reverse time equations; hence simplifying the solution of the underlying boundary value problem. We also consider infinite horizon discounted cost optimal control problems. In this case, the mean-field formulation allows one to convert the computation of the desired optimal control law into solving a pair of forward mean-field ordinary differential equations. The McKean-Pontryagin minimum principle is tested numerically for a controlled inverted pendulum, a controlled Lorenz-63 system, and a controlled Lorenz-96 system. Although the focus is on linear-quadratic control problems, the proposed methodology is extendable to more general problems including mean-field type control formulations. |
| title | On a mean-field Pontryagin minimum principle for stochastic optimal control |
| topic | Optimization and Control Numerical Analysis 35F21, 49M99, 93E20, 70H30, 70H45 |
| url | https://arxiv.org/abs/2506.10506 |