2019/02/14 by Martin Burger, Burger, Martin, René Pinnau +5 · 2 citations
Decision Sciences · Environmental Science · Mathematics · #FOS: Mathematics #Groundwater flow and contamination studies #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.1902.05339
openalex publication_date 2019/02/14 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We derive a framework to compute optimal controls for problems with states in\nthe space of probability measures. Since many optimal control problems\nconstrained by a system of ordinary differential equations (ODE) modelling\ninteracting particles converge to optimal control problems constrained by a\npartial differential equation (PDE) in the mean-field limit, it is interesting\nto have a calculus directly on the mesoscopic level of probability measures\nwhich allows us to derive the corresponding first-order optimality system. In\naddition to this new calculus, we provide relations for the resulting system to\nthe first-order optimality system derived on the particle level, and the\nfirst-order optimality system based on L2-calculus under additional\nregularity assumptions. We further justify the use of the L2-adjoint in\nnumerical simulations by establishing a link between the adjoint in the space\nof probability measures and the adjoint corresponding to L2-calculus.\nMoreover, we prove a convergence rate for the convergence of the optimal\ncontrols corresponding to the particle formulation to the optimal controls of\nthe mean-field problem as the number of particles tends to infinity.\n