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Set-valued integration and the discrete approximation of reachable sets. (Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen.) (German) Zbl 0841.65013
Bayreuther Mathematische Schriften 50. Bayreuth: Univ. Bayreuth, xxii, 248 S. (1995).
The book consists of an introduction and of four chapters. First, the definitions of some notions as the Hausdorff distance between two nonempty subsets of $$\mathbb{R}^n$$, the support function of a nonempty convex subset of $$\mathbb{R}^n$$, the averaged moduli of smoothness of a bounded function and the Aumann integral of a set-valued map are given and some of their properties are presented. Next, set-valued analogies of the Newton-Cotes formulas, of the quadrature formulas of Gauss type and Romberg formulas are presented and the Hausdorff distance between the Aumann integral of a set-valued map and the corresponding approximation is estimated.
The reachable set of a differential inclusion is defined in chapter two and composed quadrature formulas (of order one, two and greater than two) for its approximation are proposed for the case of linear differential inclusion. In chapter three algorithms are presented for numerical approximation of the reachable set of linear differential inclusions based on a direct treatment of the corresponding sets or by using of a dual approach (support functions or support points). Some numerical results are presented.

##### MSC:
 65D32 Numerical quadrature and cubature formulas 28-02 Research exposition (monographs, survey articles) pertaining to measure and integration 28A78 Hausdorff and packing measures 41A55 Approximate quadratures 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 26E25 Set-valued functions 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis