TY  - JOUR
A1  - Breitenbach, Tim
A1  - Borzì, Alfio
T1  - The Pontryagin maximum principle for solving Fokker–Planck optimal control problems
JF  - Computational Optimization and Applications
N2  - 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.
KW  - Fokker–Planck equation
KW  - Pontryagin maximum principle
KW  - non-smooth optimal control problems
KW  - stochastic processes
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-232665
SN  - 0926-6003
VL  - 76
ER  - 
TY  - JOUR
A1  - Roy, S.
A1  - Borzì, A.
A1  - Habbal, A.
T1  - Pedestrian motion modelled by Fokker-Planck Nash games
JF  - Royal Society Open Science
N2  - 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.
KW  - Fokker–Planck equation
KW  - Nash equilibrium
KW  - pedestrian motion
KW  - differential games
KW  - avoidance
KW  - optimal control
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-170395
VL  - 4
IS  - 9
ER  -