# Computing Aumann's Integral

R. Baier, F. Lempio:
**Computing Aumann's Integral**

in:
*Modeling Techniques for Uncertain Systems,Proceedings of a Conference held in
Sopron, Hungary, July 6-10, 1992,*
*
A. B. Kurzhanski, V. M. Veliov
(eds.)
*

*Progress in Systems and Control Theory*

**18**, Birkhäuser, Basel, 1994, 71 - 92

MR Nummer:
1287649

Zentralblattnummer:
0828.65021

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

Mathematics Subject Classification Code: 34A60 (49M25 65D30 65D32 65L05 93B03)

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## 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.

## Contents:

1. | Introduction |

2. | Quadrature Formulae for Set-Valued Mappings |

3. | Approximation of Reachable Sets for Linear Control Problems |

4. | Test Examples |