Journal
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
Volume 59, Issue 2, Pages 977-1006Publisher
SIAM PUBLICATIONS
DOI: 10.1137/19M1249461
Keywords
optimal control with ODE/PDE constraints; interacting particle systems; mean-field limits
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Funding
- NWO Vidi grant [016.Vidi.189.102]
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A framework for computing optimal controls for problems with states in the space of probability measures was derived. New calculus was used to derive the first-order optimality system and establish a link between the adjoint in the space of probability measures and L-2 calculus in numerical simulations. Convergence rate of optimal controls from particle formulation to mean-field problem was proven as the number of particles tends to infinity.
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.
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