Asymptotic Attainability

1997
ISBN: 978-0-7923-4302-8
Verlag: Springer Netherlands

Buch, Englisch, Band 383, 322 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 1440 g

Reihe: Mathematics and Its Applications

In this monograph, questions of extensions and relaxations are consid­ ered. These questions arise in many applied problems in connection with the operation of perturbations. In some cases, the operation of "small" per­ turbations generates "small" deviations of basis indexes; a corresponding stability takes place. In other cases, small perturbations generate spas­ modic change of a result and of solutions defining this result. These cases correspond to unstable problems. The effect of an unstability can arise in extremal problems or in other related problems. In this connection, we note the known problem of constructing the attainability domain in con­ trol theory. Of course, extremal problems and those of attainability (in abstract control theory) are connected. We exploit this connection here (see Chapter 5). However, basic attention is paid to the problem of the attainability of elements of a topological space under vanishing perturba­ tions of restrictions. The stability property is frequently missing; the world of unstable problems is of interest for us. We construct regularizing proce­ dures. However, in many cases, it is possible to establish a certain property similar to partial stability. We call this property asymptotic nonsensitivity or roughness under the perturbation of some restrictions. The given prop­ erty means the following: in the corresponding problem, it is the same if constraints are weakened in some "directions" or not. On this basis, it is possible to construct a certain classification of constraints, selecting "di­ rections of roughness" and "precision directions".

Zielgruppe

Research

Autoren/Hrsg.

Chentsov, A. G.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis
• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie
• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik
• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Mathematische Analysis Variationsrechnung
• Mathematik | Informatik Mathematik Operations Research Spieltheorie

Inhaltsverzeichnis

1 Asymptotically Attainable Elements: Model Examples.- 2 Asymptotic Effects in Linear Control Problems with Integral Constraints.- 3 Asymptotic Attainability: General Questions.- 4 Asymptotic Attainability under Perturbation of Integral Constraints.- 5 Relaxations of Extremal Problems.- 6 Some Generalizations.- 7 Other Extension Constructions in the Space of Solutions.- Conclusion.- List of notations.