Mean-field optimal control and optimality conditions in the space of probability measures

Burger M, Pinnau R, Totzeck C, Tse O (2021)

Publication Type: Journal article

Publication year: 2021

Journal

SIAM Journal on Control and Optimization Society for Industrial and Applied Mathematics

Book Volume: 59

Pages Range: 977-1006

Journal Issue: 2

DOI: 10.1137/19M1249461

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 L2-calculus under additional regularity assumptions. We further justify the use of the L2-adjoint in numerical simulations by establishing a link between the adjoint in the space of probability measures and the adjoint corresponding to L2-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.

Authors with CRIS profile

Martin Burger Lehrstuhl für Angewandte Mathematik (Modellierung und Numerik)

Involved external institutions

Technische Universität Kaiserslautern DE Germany (DE) Eindhoven University of Technology / Technische Universiteit Eindhoven (TU/e) NL Netherlands (NL)

How to cite

APA:

Burger, M., Pinnau, R., Totzeck, C., & Tse, O. (2021). Mean-field optimal control and optimality conditions in the space of probability measures. SIAM Journal on Control and Optimization, 59(2), 977-1006. https://dx.doi.org/10.1137/19M1249461

MLA:

Burger, Martin, et al. "Mean-field optimal control and optimality conditions in the space of probability measures." SIAM Journal on Control and Optimization 59.2 (2021): 977-1006.

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