TY  - JOUR
A1  - Schindele, Andreas
A1  - Borzì, Alfio
T1  - Proximal Methods for Elliptic Optimal Control Problems with Sparsity Cost Functional
JF  - Applied Mathematics
N2  - First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.
KW  - semismooth Newton method
KW  - optimal control
KW  - elliptic PDE
KW  - nonsmooth optimization
KW  - proximal method
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-145850
VL  - 7
IS  - 9
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  - 
TY  - JOUR
A1  - Karl, Veronika
A1  - Neitzel, Ira
A1  - Wachsmuth, Daniel
T1  - A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems
JF  - Computational Optimization and Applications
N2  - In this paper we apply an augmented Lagrange method to a class of semilinear ellip-tic optimal control problems with pointwise state constraints. We show strong con-vergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.
KW  - optimal control
KW  - semilinear elliptic operators
KW  - state constraints
KW  - augmented Lagrange method
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-232811
SN  - 0926-6003
VL  - 77
ER  -