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Robert Baier

Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen (Set-Valued Integration and the Discrete Approximation of Reachable Sets)

The paper is published in:
Bayreuther Mathematische Schriften 50 (1995),
Bayreuth: Universität Bayreuth, xxii + 248 S.

MSC:
65D32 Quadrature and cubature formulas
28-02 Research exposition (monographs, survey articles)
28A78 Hausdorff measures
41A55 Approximate quadratures
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41-02 Research exposition (monographs, survey articles)
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections, See Also {26E25, 54C60, 54C65, 90A14}
26E25 Set-valued functions, See also {28B20, 54C60}, {For nonsmooth analysis, See 49J52, 58Cxx, 90Cxx}
65-02 Research exposition (monographs, survey articles)

Keywords:
set-valued integration; Hausdorff distance; Aumann integral; Newton-Cotes formulas; quadrature formulas;
Romberg formulas; reachable set; differential inclusion; numerical results

Abstract:
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 Rn, the support function of a nonempty convex subset of Rn, 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.

[ Review by M.I.Krastanov (Sofia) ]

Table of Contents:
0. Hilfsmittel
0.1. Mengenoperationen und Stützfunktionen
0.1.1. Mengenoperationen
0.1.2. Hausdorff-Abstand von Mengen
0.1.3. Eigenschaften von Stützfunktionen
0.2. Variation von Funktionen und Glattheitsmoduli
1. Mengenwertige Integration
1.1. Eigenschaften des Aumann-Integrals
1.2. Konvergenzsätze für numerische Verfahren
1.3. Newton-Cotes-Verfahren
1.4. Gauß-Quadratur-Verfahren
1.5. Romberg-Verfahren
1.6. Charakterisierungen und Beispiele glatter Stützfunktionen
2. Diskrete Approximation erreichbarer Mengen linearer Differentialinklusionen
2.1. Eigenschaften erreichbarer Mengen
2.2. Quadraturverfahren zur Approximation erreichbarer Mengen
2.3. Kombinationsverfahren
2.3.1. Kombinationsverfahren der Ordnung 1
2.3.2. Kombinationsverfahren der Ordnung 2
2.3.3. Kombinationsverfahren der Ordnung größer als 2
2.3.4. Extrapolationsverfahren
3. Anwendungen
3.1. Implementierung der Algorithmen
3.1.1. Direkte Behandlung der Mengen
3.1.2. Dualer Zugang mit Stützfunktionen
3.1.3. Dualer Zugang mit Stützpunkten
3.1.4. Fehlerbetrachtung für den dualen Zugang
3.2. Beispiele zur Integration
3.3. Beispiele zur Bestimmung erreichbarer Mengen


© Robert Baier
Last modified: Thu May 28 15:11:20 MDT 1998