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Approximation of reachable sets using optimal control algorithms

URN to cite this document: urn:nbn:de:bvb:703-epub-1989-8

Title data

Baier, Robert ; Gerdts, Matthias ; Xausa, Ilaria:
Approximation of reachable sets using optimal control algorithms.
Department of Mathematics, University of Bayreuth
Bayreuth , 2012 . - 44 S.

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Project information

Project title:
Project's official titleProject's id
Marie-Curie Initial Training Network "Sensitivity Analysis for Deterministic Controller Design" (SADCO)264735-SADCO
HIM Junior Trimester Program "Computational Mathematics", Research Group "Numerical discretization methods for differential inclusions and applications to robust optimal control problems"Group C

Project financing: 7. Forschungsrahmenprogramm für Forschung, technologische Entwicklung und Demonstration der Europäischen Union
Hausdorff Research Institute for Mathematics (HIM) in Bonn


We investigate and analyze a computational method for the approximation of reachable sets for nonlinear dynamic systems. The method uses grids to cover the region of interest and the distance function to the reachable set evaluated at grid points. A convergence analysis is provided and shows the convergence of three different types of discrete set approximations to the reachable set. The distance functions can be computed numerically by suitable optimal control problems in combination with direct discretization techniques which allows adaptive calculations of reachable sets. Several numerical examples with nonconvex reachable sets are presented.

Further data

Item Type: Preprint, postprint
Additional notes (visible to public): original version from April 2010, published as technical report in October 2011, updated in October 2012


1. Introduction
2. Proximal Normals and Inner/Outer Approximation of Sets
2.1 Set Representation Techniques
2.3 Inner/Outer Approximation of Sets
3. Convergence Analysis
3.1 Properties and Approximations of Reachable Sets
3.2 Discrete Approximation of Reachable Sets
4. Numerical Realization
4.1 DFOG Method
5. Numerical Examples
5.1 Kenderov's Example
5.2 Bilinear Example
5.3 Adaptive Version
5.4 Example from a Pursuit-Evasion Game
6. Outline
Keywords: reachable sets; optimal control; direct discretization
Subject classification: Mathematics Subject Classification Code: 49J15 49M25 93B03 93C10 (90C30)
DDC Subjects: 500 Science > 510 Mathematics
Institutions of the University: Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Profile Fields
Profile Fields > Advanced Fields
Profile Fields > Advanced Fields > Nonlinear Dynamics
Language: English
Originates at UBT: Yes
URN: urn:nbn:de:bvb:703-epub-1989-8
Date Deposited: 10 Apr 2015 06:36
Last Modified: 28 Mar 2019 15:38
URI: https://epub.uni-bayreuth.de/id/eprint/1989

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