@article{BreitenbachBorzi2020,
  author    = {Breitenbach, Tim and Borz{\`i}, Alfio},
  title     = {The Pontryagin maximum principle for solving Fokker-Planck optimal control problems},
  series = {Computational Optimization and Applications},
  volume    = {76},
  journal   = {Computational Optimization and Applications},
  issn      = {0926-6003},
  doi       = {10.1007/s10589-020-00187-x},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232665},
  pages     = {499-533},
  year      = {2020},
  abstract  = {The characterization and numerical solution of two non-smooth optimal control problems governed by a Fokker-Planck (FP) equation are investigated in the framework of the Pontryagin maximum principle (PMP). The two FP control problems are related to the problem of determining open- and closed-loop controls for a stochastic process whose probability density function is modelled by the FP equation. In both cases, existence and PMP characterisation of optimal controls are proved, and PMP-based numerical optimization schemes are implemented that solve the PMP optimality conditions to determine the controls sought. Results of experiments are presented that successfully validate the proposed computational framework and allow to compare the two control strategies.},
  language  = {en}
}
@article{RoyBorziHabbal2017,
  author    = {Roy, S. and Borz{\`i}, A. and Habbal, A.},
  title     = {Pedestrian motion modelled by Fokker-Planck Nash games},
  series = {Royal Society Open Science},
  volume    = {4},
  journal   = {Royal Society Open Science},
  number    = {9},
  doi       = {10.1098/rsos.170648},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-170395},
  pages     = {170648},
  year      = {2017},
  abstract  = {A new approach to modelling pedestrians' avoidance dynamics based on a Fokker-Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to the output of real experiments (conducted with humans) for four test cases.},
  language  = {en}
}