In the framework of Baoding Liu’s uncertainty theory, some new concepts are introduced and their properties are considered. In particular, regular functions of uncertainty are introduced on an uncountable product of spaces. An analog of the Lomnitskii–Ulam theorem from traditional probability theory is obtained. Necessary and sufficient conditions are specified under which a function defined on a Banach space of bounded functions is a distribution function for some uncertain mapping. Some notions of Liu’s theory are generalized for uncountably many objects. Examples showing the similarity and the difference between Liu’s theory and probability theory are analyzed. An application of Liu’s theory to estimation theory is considered on examples.
Keywords: functions of uncertainty, uncertain mappings, distribution functions, theory of estimation
Received July 15, 2023
Revised October 20, 2023
Accepted October 23, 2023
Funding Agency: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2023-913).
Boris Ivanovich Ananyev, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: abi@imm.uran.ru
REFERENCES
1. Liu Baoding. Uncertainty theory, Ser. Springer Uncertainty Research, 2015, 487 p. doi: 10.1007/978-3-662-44354-5
2. Yao Kai. Uncertain differential equations, Ser. Springer Uncertainty Research, 2016, 158 p. doi: 10.1007/978-3-662-52729-0
3. Zhu Yuanguo. Uncertain optimal control. Ser. Springer Uncertainty Research, 2019, 208 p. doi: 10.1007/978-981-13-2134-4
4. Sheng L., Zhu Y., Yan H., Wang K. Uncertain optimal control approach for CO2 mitigation problem. Asian J. Control, 2017, vol. 19, no. 6, pp. 1931–1942. doi: 10.1002/asjc.1524
5. Hou Yongchao, Yang Shengyan. Uncertain programming with recourse. J. Chem. Pharm. Research, 2015, vol. 7, no. 3, pp. 2366–2372.
6. Liu Baoding. Toward uncertain finance theory. J. Uncertainty Anal. Appl., 2013, vol. 1, art. no. 1. doi: 10.1186/2195-5468-1-1
7. Ananyev B.I. A state estimation of Liu equations. AIP Conference Proceedings, 2015, vol. 1690, art. no. 040005. doi: 10.1063/1.4936712
8. Kaufmann A. Introduction to the theory of fuzzy subsets, NY, Acad. Press, 1975, 416 p. ISBN: 0124023010 . Translated to Russian under the title Vvedenie v teoriyu nechetkikh mnozhestv, Moscow, Radio i Svyaz’ Publ., 1982, 432 p.
9. Molodtsov D.A. Teoriya myagkikh mnozhestv [Theory of soft sets], Moscow, Editorial URSS, 2004. ISBN: 5-354-00810-7 .
10. Malinetskii G.G. Khaos. Struktury. Vychislitel’nyi eksperiment. Vvedenie v nelineinuyu dinamiku [Chaos. Structures. Computational experiment. Introduction to nonlinear dynamics]. Moscow, Editorial URSS, 2001, 256 p. ISBN: 5-354-00063-7 .
11. Pyt’ev Yu.P. Vozmozhnost’. Elementy teorii i primeneniya [Possibility. Elements of theory and application]. Moscow, Editorial URSS, 2000, 192 p. ISBN: 5-8360-0129-4 .
12. Pyt’ev Yu.P. Vozmozhnost’ kak al’ternativa veroyatnosti. Matematicheskie i empiricheskie osnovy, prilozheniya [Possibility as an alternative to probability. Mathematical and empirical fundamentals, applications]. Moscow, Fizmatlit Publ., 2016, 600 p. ISBN: 978-5-9221-1657-2 .
13. Kurzhanskii A.B. Upravlenie i nabludenie v usloviyakh neopredelennosti [Control and observation in conditions of indefiniteness]. Moscow, Nauka Publ., 1977, 392 p.
14. Kurzhanski A., Varaiya P. Dynamics and control of trajectory tubes: theory and computation. Ser. Systems & Control: Foundations & Applications, Boston, Birkhäuser, 2014, 445 p. doi: 10.1007/978-3-319-10277-1
15. Shiryaev A.N. Veroyatnost’ 1, 2 [Probability 1, 2]. Moscow, Mosk. Tsentr Nepr. Matem. Obr. Publ., 2004, 924 p. ISBN: 5-94057-036-4 .
16. Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov [The theory of stochastic processes]. Moscow, Fizmatlit Publ., 2003, 399 p. ISBN: 5-9221-0335-0 .
Cite this article as: B.I. Ananyev. On some complements to Liu’s theory. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 5–20. Proceedings of the Steklov Institute of Mathematics, 2024, Vol. 325, Suppl. 1, pp. S1–S16.
