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Optimization Problems for Interacting Particle Systems and Corresponding Mean‐field Limits
Author(s) -
Pinnau René,
Totzeck Claudia
Publication year - 2019
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201900148
Subject(s) - limit (mathematics) , metric (unit) , particle system , space (punctuation) , mathematics , wasserstein metric , field (mathematics) , metric space , kinetic energy , statistical physics , relation (database) , mathematical optimization , pure mathematics , computer science , physics , mathematical analysis , classical mechanics , operations management , economics , operating system , database
We summarize the relations of optimality systems for an interacting particle dynamic in the microscopic and in the kinetic description. In particular, we answer the question if the passing to the mean‐field limit and deriving the first order optimality system can be interchanged without affecting the results. The answer is affirmative, if one derives the optimality system on the kinetic level in the metric space ( 2 , 2 ). Moreover, we discuss the relation of to the adjoint PDE derived in the L 2 ‐sense. Here, the gradient can be derived as expected from the calculus in Wasserstein space.