Burger M, Pinnau R, Totzeck C, Tse O (2021)
Publication Type: Journal article
Publication year: 2021
Book Volume: 59
Pages Range: 977-1006
Journal Issue: 2
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.
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
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.