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Analytical and numerical estimates of reachable sets in a subdivision scheme

Title data

Riedl, Wolfgang ; Baier, Robert ; Gerdts, Matthias:
Analytical and numerical estimates of reachable sets in a subdivision scheme.
Mathematisches Institut, Universität Bayreuth; Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr in Neubiberg/München
Bayreuth, Neubiberg/München , 2017 . - 15 p.
DOI: https://doi.org/10.15495/EPub_UBT_00005055

Official URL: Volltext

Abstract in another language

Reachable sets for (discrete) nonlinear control problems can be described by feasible sets of nonlinear optimization problems. The objective function for this problem is set to minimize the distance from an arbitrary grid point of a bounding box to the reachable set.

To avoid the high computational costs of starting the optimizer for all points in an equidistant grid, an adaptive version based on the subdivision framework known in the computation of attractors and invariant measures is studied. The generated box collections provide over-approximations which shrink to the reachable set for a decreasing maximal diameter of the boxes in the collection and, if the bounding box is too pessimistic, do not lead to an exploding number of boxes as examples show. Analytical approaches for the bounding box of a 3d funnel are gained via the Gronwall-Filippov-Wazewski theorem for differential inclusions or by choosing good reference solutions. An alternative self-finding algorithm for the bounding box is applied to a higher-dimensional kinematic car model.

Further data

Item Type: Preprint, postprint
Additional notes: Contents:
1. Introduction
1.1 Reachability analysis
1.2 Preliminaries
1.3 Control problems and differential inclusions
1.4 Direct discretization via set-valued Runge-Kutta methods
2. Subdivision Algorithm for Reachable Sets and Its Convergence
2.1 Non-adaptive and adaptive algorithm
2.2 Convergence study
3. Analytical and Numerical Calculation of Bounding Boxes
3.1 Analytical approach
3.2 Numerical approach
4. Examples
4.1 Kenderov’s example
4.2 Car model
5. Conclusions
Keywords: reachable sets; subdivision; direct discretization of optimal control;
Filippov's theorem; nonlinear optimization
Subject classification: Mathematics Subject Classification Code: 93B03 34A60 (49M25 49J53 65L07 93D23 93C10)
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)
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) > Chair Mathematics V (Applied Mathematics) - Univ.-Prof. Dr. Lars Grüne
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Scientific Computing
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Scientific Computing > Chair Scientific Computing - Univ.-Prof. Dr. Mario Bebendorf
Profile Fields
Profile Fields > Advanced Fields
Profile Fields > Advanced Fields > Nonlinear Dynamics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 19 Sep 2020 21:00
Last Modified: 29 Sep 2020 10:02
URI: https://eref.uni-bayreuth.de/id/eprint/57313