Geodesic
On some complements to Liu's theory
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 5-20
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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. 
@article{TIMM_2024_30_1_a0,
     author = {B. I. Anan'ev},
     title = {On some complements to {Liu's} theory},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {5--20},
     year = {2024},
     volume = {30},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a0/}
}
TY  - JOUR
AU  - B. I. Anan'ev
TI  - On some complements to Liu's theory
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2024
SP  - 5
EP  - 20
VL  - 30
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a0/
LA  - ru
ID  - TIMM_2024_30_1_a0
ER  - 
%0 Journal Article
%A B. I. Anan'ev
%T On some complements to Liu's theory
%J Trudy Instituta matematiki i mehaniki
%D 2024
%P 5-20
%V 30
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a0/
%G ru
%F TIMM_2024_30_1_a0
B. I. Anan'ev. On some complements to Liu's theory. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 5-20. http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a0/

[1] Baoding Liu, Uncertainty theory, Springer Uncertainty Research, 4-th ed., 2015, 487 pp. | DOI | MR | Zbl

[2] Kai Yao, Uncertain differential equations, Springer Uncertainty Research, 2016, 158 pp. | DOI | MR | Zbl

[3] Zhu Yuanguo, Uncertain optimal control, Springer Uncertainty Research, 2019, 208 pp. | DOI | MR

[4] Sheng L., Zhu Y., Yan H., Wang K., “Uncertain optimal control approach for CO2 mitigation problem”, Asian J. Control, 19:6 (2017), 1931–1942 | DOI | MR | Zbl

[5] Hou Yongchao, Yang Shengyan, “Uncertain programming with recourse”, J. Chem. Pharm. Research, 7:3 (2015), 2366–2372

[6] Baoding Liu, “Toward uncertain finance theory”, J. Uncertainty Anal. Appl., 1 (2013), 1, 15 pp. | DOI | MR | Zbl

[7] Ananyev B.I., “A State Estimation of Liu Equations”, AIP Conference Proceedings, 1690, 2015, 040005 | DOI

[8] Kofman A., Vvedenie v teoriyu nechetkikh mnozhestv, Radio i svyaz, M., 1982, 432 pp.

[9] Molodtsov D.A., Teoriya myagkikh mnozhestv, URSS, M., 2004

[10] Malinetskii G.G., Khaos. Struktury. Vychislitelnyi eksperiment. Vvedenie v nelineinuyu dinamiku, 3-e izd., URSS, M., 2001, 256 pp.

[11] Pytev Yu.P., Vozmozhnost. Elementy teorii i primeneniya, URSS, M., 2000, 192 pp.

[12] Pytev Yu.P., Vozmozhnost kak alternativa veroyatnosti. Matematicheskie i empiricheskie osnovy, prilozheniya, 2-e izd., pererab. i dop., Fizmatlit, M., 2016, 600 pp.

[13] Kurzhanskii A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 392 pp.

[14] Kurzhanski A., Varaiya P., Dynamics and control of trajectory tubes: Theory and computation, SCFA, Birkhäuser, Boston, 2014, 445 pp. | DOI | MR | Zbl

[15] Shiryaev A.N.,, Veroyatnost 1, 2, Izd-vo MTsNMO, M., 2004, 924 pp.

[16] Bulinskii A.V., Shiryaev A.N., Teoriya sluchainykh protsessov, Fizmatlit, M., 2003, 399 pp.