Section
|
Mathematics
|
Title
|
Unlocking of predicate: application to constructing a non-anticipating selection
|
Author(-s)
|
Serkov D.A.ab
|
Affiliations
|
Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa,
Ural Federal Universityb
|
Abstract
|
We consider an approach to constructing a non-anticipating selection of a multivalued mapping; such a problem arises in control theory under conditions of uncertainty. The approach is called “unlocking of predicate” and consists in the reduction of finding the truth set of a predicate to searching fixed points of some mappings. Unlocking of predicate gives an extra opportunity to analyze the truth set and to build its elements with desired properties. In this article, we outline how to build “unlocking mappings” for some general types of predicates: we give a formal definition of the predicate unlocking operation, the rules for the construction and calculation of “unlocking mappings” and their basic properties. As an illustration, we routinely construct two unlocking mappings for the predicate “be non-anticipating mapping” and then on this base we provide the expression for the greatest non-anticipating selection of a given multifunction.
|
Keywords
|
predicate unlocking, fixed points, nonanticipating mappings
|
UDC
|
510.635, 517.988.52, 519.833, 517.977
|
MSC
|
37N35, 65J15, 47J25, 52A01, 91A25
|
DOI
|
10.20537/vm170211
|
Received
|
1 February 2017
|
Language
|
English
|
Citation
|
Serkov D.A. Unlocking of predicate: application to constructing a non-anticipating selection, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 2, pp. 283-291.
|
References
|
- Kakutani S. A generalization of Brouwer's fixed point theorem, Duke Math. J., 1941, vol. 8, no. 3, pp. 457-459. DOI: 10.1215/S0012-7094-41-00838-4
- Nash J. Non-cooperative games, Ann. of Math., 1951, vol. 54, no. 2, pp. 286-295. DOI: 10.2307/1969529
- Nikaido H. On von Neumann's minimax theorem, Pacific Journal of Mathematics, 1954, vol. 4, no. 1, pp. 65-72. DOI: 10.2140/pjm.1954.4.65
- Chentsov A.G. On the structure of a game problem of convergence, Sov. Math., Dokl., 1975, vol. 16, no. 5, pp. 1404-1408.
- Chentsov A.G. On a game problem of guidance, Sov. Math., Dokl., 1976, vol. 17, pp. 73-77.
- Chentsov A.G. Non-anticipating selectors for set-valued mappings, Differ. Uravn. Protsessy Upr., 1998, no. 2, pp. 29-64 (in Russian).
- Chentsov A.G. Hereditary multiselectors of set-valued mappings and their constructing by iteration methods, Differ. Uravn. Protsessy Upr., 1999, no. 3, pp. 1-54 (in Russian).
- Serkov D.A. An approach to analysis of the set of truth: unlocking of predicate, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2016, vol. 26, issue 4, pp. 525-534 (in Russian). DOI: 10.20537/vm160407
- Kuratowski K., Mostowski A. Set theory, Amsterdam: North-Holland, 1967, xi+417 p. Translated under the title Teoriya mnozhestv, Moscow: Mir, 1970, 416 p.
- Markowsky G. Chain-complete posets and directed sets with applications, Algebra Universalis, 1976, vol. 6, issue 1, pp. 53-68. DOI: 10.1007/BF02485815
- Serkov D.A. Transfinite sequences in the method of programmed iterations, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2017, vol. 23, no. 1, pp. 228-240 (in Russian). DOI: 10.21538/0134-4889-2017-23-1-228-240
- Tarski A. A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics, 1955, vol. 5, no. 2, pp. 285-309. DOI: 10.2140/pjm.1955.5.285
- Cousot P., Cousot R. Constructive versions of Tarski’s fixed point theorems, Pacific Journal of Mathematics, 1979, vol. 82, no. 1, pp. 43-57. http://projecteuclid.org/euclid.pjm/1102785059
- Fan K. Minimax theorems, Proc. Natl. Acad. Sci. USA, 1953, vol. 39, no. 1, pp. 42-47.
- Serkov D.A. On fixed point theory and its applications to equilibrium models, Bulletin of the South Ural State University, Series Mathematical Modelling, Programming and Computer Software, 2016, vol. 9, no. 1, pp. 20-31. DOI: 10.14529/mmp160102
|
Full text
|
|