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 | |