# Set-Valued 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 Baier### project 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 right-hand sides, dynamical systems with uncertainties, non-smooth optimization problems, and optimal control problems lead to differential equations with set-valued right-hand sides, so-called differential inclusions.

Within the research project "Set-Valued 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 set-valued numerical analysis, especially integration techniques for set-valued integrands and interpolation techniques for set-valued maps, based e.g. on set-valued finite element methods, set-valued 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.

## 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**

Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen

Bayreuth, Universität, 1995(dissertation, , )

**Article in a journal (assessed)**

Regularity of set-valued maps and their selections through set differences. Part 1: Lipschitz c ...

*in*Serdica Mathematical Journal volume 39 (2013) issue 3-4. - page 365-390

http://www.math.bas.bg/serdica/2013/2013-365-390.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 39-69

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 519-548

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 323-330

http://www.proceedings.bas.bg/content/2014_3_cnten ...

rev.: AMS MathSciNet

revZentralblatt

Regularity of set-valued maps and their selections through set differences. Part 2: One-sided L ...

*in*Serdica Mathematical Journal volume 39 (2013) issue 3-4. - page 391-422

http://www.math.bas.bg/serdica/2013/2013-391-422.p ...

rev.: AMS MathSciNet

revZentralblatt

**Article in a book (assessed)**

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 9-13, 2012, Melbourne, Australia

Melbourne, Australia : Think Business Events, 2012. - page 8

http://www.mtns2012.conference.net.au/Full%20Paper ...

**Conference item (not assessed)**

Software for verification of collision avoidance algorithms via optimal control techniques

2014

https://hal.inria.fr/hal-01024587

Conference on New Trends in Optimal Control (NetCO 2014), Tours, Frankreich

**Preprint, postprint, working paper, discussion paper**

Construction of the Minimum Time Function Via Reachable Sets of Linear Control Systems. Part 1: ...

Bayreuth ; Padova, 2015. - 30 page

Optimization-based subdivision algorithm for reachable sets

Bayreuth, 2016. - 33 page

http://nbn-resolving.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 Hamilton-Jacobi approach

Paris, France, 2014. - 22 page

https://hal.inria.fr/hal-01123490

## publications

Please go to the list of publications of this project.