Abstract

We derive a framework to compute optimal controls for problems with states in the space of probability measures. Since many optimal control problems constrained by a system of ordinary differential equations modeling interacting particles converge to optimal control problems constrained by a partial differential equation in the mean-field limit, it is interesting to have a calculus directly on the mesoscopic level of probability measures which allows us to derive the corresponding first-order optimality system. In addition to this new calculus, we provide relations for the resulting system to the first-order optimality system derived on the particle level and the first-order optimality system based on $L^2$-calculus under additional regularity assumptions. We further justify the use of the $L^2$-adjoint in numerical simulations by establishing a link between the adjoint in the space of probability measures and the adjoint corresponding to $L^2$-calculus. Moreover, we prove a convergence rate for the convergence of the optimal controls corresponding to the particle formulation to the optimal controls of the mean-field problem as the number of particles tends to infinity.

Keywords

  1. optimal control with ODE/PDE constraints
  2. interacting particle systems
  3. mean-field limits

MSC codes

  1. 49K15
  2. 49K20

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Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 977 - 1006
ISSN (online): 1095-7138

History

Submitted: 12 March 2019
Accepted: 28 December 2020
Published online: 9 March 2021

Keywords

  1. optimal control with ODE/PDE constraints
  2. interacting particle systems
  3. mean-field limits

MSC codes

  1. 49K15
  2. 49K20

Authors

Affiliations

Funding Information

Nederlandse Organisatie voor Wetenschappelijk Onderzoek https://doi.org/10.13039/501100003246 : 016.Vidi.189.102

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