SetValued Numerical Analysis
start of the project: 2008
funding institution: The Hermann Minkowski Center for Geometry in Tel Aviv, Israel; DAAD; Bulgarian Academy of Sciences
project members
principal investigator
AD Dr. Robert Baierproject members
Dr. Gilbert Perria (project member until 09/2010 )
aims of the project
The project leader continues the project which was started 1995 by Prof. Dr. Frank Lempio.
Modelling of differential equations with discontinuous righthand sides, dynamical systems with uncertainties, nonsmooth optimization problems, and optimal control problems lead to differential equations with setvalued righthand sides, socalled differential inclusions.
Within the research project "SetValued Numerical Analysis", differential inclusions are investigated qualitatively and quantitatively, and algorithms for their numerical solution are developed.
The numerical solution of such differential inclusions requires algorithms from setvalued numerical analysis, especially integration techniques for setvalued integrands and interpolation techniques for setvalued maps, based e.g. on setvalued finite element methods, setvalued divided differences and embeddings of convex compact sets. Especially, one focus is laid on the approximation of reachable sets at a given time which consist of all endpoints of admissible trajectories of a differential inclusion.
There are strong connections to the direcet discretization of optimal control problems, e.g. in questions on the (order of) convergence of the optimal control/trajectory of the discrete problems to the corresponding optimal control/trajectory of the continuous problem.
Minisymposia/Sessions
The project members organized the following minisymposia and sessions on setvalued numerics.

invited Special Session "Approximation of Differential Inclusions and Control Problems 12"
conference: 14th Viennese Conference on Optimal Control and Dynamic Games (OCDGND 2017) in Wien, Austria
chair: Robert Baier, Elza Farkhi (Tel Aviv University, Israel)

invited Special Session "SetValued Numerics and Control 12"
conference: 13th Viennese Workshop on Optimal Control, and Dynamic Games (OCDGND 2015) in Wien, Austria
chair: Robert Baier, Elza Farkhi (Tel Aviv University, Israel)

invited Special Session "SetValued Analysis for Optimal Control 12"
conference: 12th Viennese Workshop on Optimal Control, Dynamic Games and Nonlinear Dynamics (OCDGND 2012) in Wien, Austria
chair: Robert Baier
Links
 The Hermann Minkowski Center of Geometry (Minerva Center at the Tel Aviv University)
 Minerva Foundation
Selected publications of the project at the ERef server
Doctoral thesis
Baier, RobertMengenwertige Integration und die diskrete Approximation erreichbarer Mengen
Bayreuth, Universität, 1995(dissertation, , )
Article in a journal (not assessed)
Riedl, WolfgangOptimizationbased subdivision algorithm for reachable sets
in Proceedings in Applied Mathematics and Mechanics volume 14 (2014) issue 1.  page 937938
doi:10.1002/pamm.201410449 ...
85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Erlangen 2014, Erlangen, Germany
Article in a journal (assessed)
Baier, Robert; Farkhi, ElzaRegularity of setvalued maps and their selections through set differences. Part 1: Lipschitz c ...
in Serdica Mathematical Journal volume 39 (2013) issue 34.  page 365390
http://www.math.bas.bg/serdica/2013/2013365390.p ...
rev.: AMS MathSciNet
revZentralblatt
The computation of convex invariant sets via Newton's method
in Journal of Computational Dynamics volume 1 (2014) issue 1.  page 3969
doi:10.3934/jcd.2014.1.39 ...
rev.: Zentralblatt
revAMS MathSciNet
Approximation of reachable sets using optimal control algorithms
in Numerical Algebra, Control and Optimization volume 3 (2013) issue 3.  page 519548
doi:10.3934/naco.2013.3.519 ...
rev.: AMS MathSciNet
revZentralblatt
Existence of Solutions for Nonconvex Differential Inclusions of Monotone Type
in Comptes Rendus de l'Académie Bulgare des Sciences volume 67 (2014) issue 3.  page 323330
http://www.proceedings.bas.bg/content/2014_3_cnten ...
rev.: AMS MathSciNet
revZentralblatt
Regularity of setvalued maps and their selections through set differences. Part 2: Onesided L ...
in Serdica Mathematical Journal volume 39 (2013) issue 34.  page 391422
http://www.math.bas.bg/serdica/2013/2013391422.p ...
rev.: AMS MathSciNet
revZentralblatt
Article in a book (assessed)
Zanon, Mario; Boccia, Andrea; Palma, Vryan Gil; Parenti, Sonja; Xausa, IlariaDirect Optimal Control and Model Predictive Control
Optimal Control : Novel Directions and Applications
Cham, Switzerland : Springer, 2017.  page 263382 .  (Lecture Notes in Mathematics; 2180)
doi:10.1007/9783319607719_3 ...
rev.: Zentralblatt Mathematik
Avoidance trajectories for driver assistance systems via solvers for optimal control problems
Proceedings on the 20th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2012), July 913, 2012, Melbourne, Australia
Melbourne, Australia : Think Business Events, 2012.  page 8
http://www.mtns2012.conference.net.au/Full%20Paper ...
Conference item (not assessed)
Xausa, Ilaria; Baier, Robert; Bokanowski, Olivier; Gerdts, Matthias; Töpfer, DanielSoftware for verification of collision avoidance algorithms via optimal control techniques
2014
https://hal.inria.fr/hal01024587
Conference on New Trends in Optimal Control (NetCO 2014), Tours, Frankreich
Preprint, postprint, working paper, discussion paper
Baier, Robert; Le, Thuy Thi ThienConstruction of the Minimum Time Function Via Reachable Sets of Linear Control Systems. Part 1: ...
Bayreuth ; Padova, 2015.  30 page
Optimizationbased subdivision algorithm for reachable sets
Bayreuth, 2016.  33 page
http://nbnresolving.de/urn/resolver.pl?urn=urn:nb ...
Construction of the Minimum Time Function Via Reachable Sets of Linear Control Systems. Part 2: ...
Bayreuth ; Padova, 2015.  16 page
Computation of safety regions for driver assistance systems by using a HamiltonJacobi approach
Paris, France, 2014.  22 page
https://hal.inria.fr/hal01123490
publications
Please go to the list of publications of this project.