Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen

R. Baier: Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen
Bayreuther Mathematische Schriften 50, xxii+248 pp., 1995

ISBN/ISSN/ISMV Nummer: 0172-1062
MR Nummer: 1340285
Zentralblattnummer: 0841.65013
Keywords: set-valued integration; reachable set; Aumann integral; Newton-Cotes formulas; Romberg integration; quadrature formulas; differential inclusion
Mathematics Subject Classification Code: 93B05 (49M25 65K10); 65D32 (26E25 28-02 28A78 28B20 41-02 41A55 41A65 65-02)


Abstract:

This monograph is devoted to numerical integration of set-valued mappings. The approach chosen by the author is based on the following two facts: (1) the Aumann integral of a set-valued mapping F: [0,T] -> |Rn is a convex set and can be identified with its support function; (2) the value of the support function in any direction equals the integral of the support function of F in the same directions. For calculation of the latter integrals one can use any quadrature formula for (single-valued) functions. The accuracy of approximation depends on the chosen quadrature formula and on the ``smoothness" properties of the support function.
The realization of this approach and the corresponding error analysis are based on certain mathematical techniques that are comprehensively presented in the monograph: calculus with sets, elements of convex analysis (in particular, properties of the support function), moduli of smoothness, properties of the Aumann integral, and error estimates for classical quadrature formulae (Newton-Cotes, Gauss, Romberg, etc.).
The approach leads to a variety of set-valued quadrature formulae and corresponding error estimates. A separate chapter is devoted to numerical approximation of the reachable set of linear control systems, where error estimates are also obtained. Finally, possible computer implementations are discussed and a number of examples and computer plots are provided.
The monograph is clearly written, self-contained and could be useful for mathematicians interested in numerical analysis, control theory, set-valued analysis and differential inclusions.

Contents:

0. Hilfsmittel
0.1 Operationen von Mengen und Stützfunktionen
0.2 Variation von Funktionen und Glattheitsmoduli
2. Mengenwertige Integration und numerische Verfahren
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. Berechnung erreichbarer Mengen linearer Differentialinklusionen
2.1 Eigenschaften erreichbarer Mengen
2.2 Quadraturverfahren zur Berechnung der erreichbaren Mengen
2.3 Kombinationsverfahren
3. Anwendungen
3.1 Implementierung der Algorithmen
3.2 Beispiele zur Integration
3.3 Beispiele zur Bestimmung erreichbarer Mengen

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