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Construction of the Minimum Time Function for Linear Systems Via Higher-Order Set-Valued Methods

Title data

Baier, Robert ; Le, Thuy Thi Thien:
Construction of the Minimum Time Function for Linear Systems Via Higher-Order Set-Valued Methods.
In: Mathematical Control and Related Fields. Vol. 9 (2019) Issue 2 . - pp. 223-255.
ISSN 2156-8472
DOI: https://doi.org/10.3934/mcrf.2019012

Project information

Project title:
Project's official titleProject's id
PhD fellowship for foreign students at the Università di PadovaNo information

Project financing: Andere
Fondazione CARIPARO

Abstract in another language

The paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Higher-order discretization of the reachable set of the linear control problem can balance missing regularity (e.g., if only Hölder continuity holds) of the minimum time function for smoother problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e., we study cases in which the minimum time function is Hölder continuous or even discontinuous.

Further data

Item Type: Article in a journal
Refereed: Yes
Additional notes: Online First since Nov 2018
Contents:
1. Introduction
2. Preliminaries
3. Approximation of the minimum time function
3.1 Set-valued discretization methods
3.2 Implementation and error estimate of the reachable set approximation
3.3 Error estimate of the minimum time function
4. Convergence and reconstruction of discrete optimal controls
5. Numerical tests
5.1 Linear examples
5.2 A nonlinear example
5.3 Non-strict expanding property of reachable sets
5.4 Problematic examples
6. Conclusions
published in two parts in arXiv at December 2015
Keywords: minimum time function; reachable sets; linear control problems; higher-order set-valued methods; direct discretization methods
Subject classification: Mathematics Subject Classification Code: 49N60 93B03 (49N05 49M25 52A27)
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
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 31 Oct 2018 07:30
Last Modified: 01 Oct 2019 09:00
URI: https://eref.uni-bayreuth.de/id/eprint/46178