Computing Aumann’s integral. (English) Zbl 0828.65021

Baier, R.; Lempio, F.

Kurzhanski, Alexander B. et al., Modeling techniques for uncertain systems. Proceedings of a conference held in Sopron, Hungary, July 6-10, 1992. Boston, MA: Birkhäuser. Prog. Syst. Control Theory. 18, 71-92 (1994).

Summary: Quadrature formulae for the numerical approximation of R. J. Aumann’s integral [J. Math. Anal. Appl. 12, 1-12 (1965; Zbl 0163.063)] 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 of the numerical approximation of the support functional of Aumann’s integral by ordinary quadrature formulae. For set-valued integrands which are smooth in an approximate 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.
For the entire collection see [Zbl 0790.00014].

MSC:

65D32	Numerical quadrature and cubature formulas
65L05	Numerical methods for initial value problems involving ordinary differential equations
41A55	Approximate quadratures
65L12	Finite difference and finite volume methods for ordinary differential equations
34A60	Ordinary differential inclusions
93B05	Controllability

Keywords:

quadrature formula; Aumann’s integral; reachable set; finite difference methods; linear differential inclusions; Newton-Cotes formulae; Romberg integration; set-valued integrands; linear control problems

Citations:

Zbl 0163.063