Asymptotic Attainability

Asymptotic Attainability

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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".
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Product details

  • Hardback | 322 pages
  • 160.02 x 236.22 x 27.94mm | 589.67g
  • Dordrecht, Netherlands
  • English
  • 1997 ed.
  • XIV, 322 p.
  • 0792343026
  • 9780792343028

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This book deals with the construction of correct extensions of extremal problems including problems of multicriterial optimization and more general problems of optimization with respect to a cone. The methods of qualitative stability and asymptotically insensitive analysis proposed here are particularly applicable to problems of optimal control with integrally constrained openloop controls. A nontraditional mathematical tool using elements of finitely-additive measure theory is applied, which necessitated special research concerned with approximative analogues of the Radon-Nikodym property. These abstract constructions do, however, address the essence of the problem at hand, and may find other applications as well. This volume will be useful to specialists and graduate students whose fields of interest include control theory and its applications, measure integration, functional analysis, optimal control, fuzzy sets and fuzzy logic, and general topology.
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Table of contents

Preface. 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. Index. List of Notations.
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