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Robert Baier, Frank Lempio

Computing Aumann's Integral

The paper is published in:
Modeling Techniques for Uncertain Systems.
Proceedings of a Conference held in Sopron, Hungary, July 6-10 1992.
Alexander B. Kurzhanksi, Vladimir V. Veliov (eds.), Progress in Systems and Control Theory 18.
Birkhäuser: Boston-Basel-Berlin, 1994, pp. 71-93.

MSC:
34A60 Equations with multivalued right-hand sides, See Also
49M25 Finite difference methods
65D30 Numerical integration
65L05 Initial value problems
93B03 Attainable sets

Keywords:
Aumann's integral, reachable set, finite difference methods

Abstract:
Quadrature formulae for the numerical approximation of Aumann's integral are investigated, which are set-valued
analogues of ordinary quadrature formulae with nonnegative weights, like certain Newton-Cotes formulae or Romberg
integration.

Essentially, the approach consists in the numerical approximation of the support functional of Aumann's integral by ordinary
quadrature formulae. For set-valued integrands which are smooth in an appropriate sense, this approach yields higher order
methods, for set-valued integrands which are not smooth enough, it yields further insight into well-known order reduction
phenomena.

The results are used to define higher order methods for the approximation of reachable sets of certain classes of linear
control problems.

Table of Contents:
1. Introduction
2. Quadrature Formulae for Set-Valued Mappings
3. Approximation of Reachable Sets for Linear Control Systems
4. Test Examples


© Robert Baier
Last modified: Thu May 28 14:47:02 MDT 1998