O.V. Druzhinina - Dr.Sc. (Phys.-Math.), Professor, General Research Scientist, Federal Research Center "Informatics and Management" RAS E-mail: ovdruzh@mail.ru O.N. Masina - Dr.Sc. (Phys.-Math.), Head of Department of Mathematical Modeling and Computer Technologies, Yelets state university n.a. I. A. Bunin E-mail: olga121@inbox.ru A.A. Petrov - Post-graduate Student, Yelets State University n.a. I. A. Bunin E-mail: xeal91@yandex.ru

The technical systems modeled by differential inclusions are considered. The approaches to optimal control search of the motion of specified systems are analyzed. The first approach is connected with construction of the dynamical model in two-dimensional space taking into account the multivalence and criteria of an optimality. The second approach is the development of the first approach and second approach is based on the construction of model in three-dimensional space taking into account the multivalence and variability. The third approach is the generalization of the first and second approaches in case when air resistance is taken into account. The generalized dynamic model is offered based on the third approach, stability analysis is carried out and the algorithm with using of artificial neural networks is developed that allows to find optimal parameters of the motion in case when the analytical research of model is difficult. The results of the work can find application in the problems of construction and research of the mathematical models of the motion for technical systems, in particular, in the control problems of aircrafts motion.

 

1. Ivanov A. A., Torokhov S. L. Upravlenie v tekhnicheskikh sistemakh. M.: Forum. 2012.
2. Masina O.N. Voprosy upravlenija dvizheniem transportnykh sistem // Transport: nauka, tekhnika, upravlenie. 2006. № 12. S. 10-12.
3. SHestakov A.A., Druzhinina O.V., Masina O.N. Ocenka bezopasnosti dvizhenija relsovykh ehkipazhejj na osnove obobshhennojj tekhnicheskojj ustojjchivosti i ustojjchivosti po ZHukovskomu // Transport: Nauka, tekhnika, upravlenie. 2014. № 2. S. 3-8.
4. Masina O.N., Druzhinina O.V. Modelirovanie i analiz ustojjchivosti nekotorykh klassov sistem upravlenija. M.: VC RAN. 2011.
5. Druzhinina O.V., Masina O.N. Metody analiza ustojjchivosti dinamicheskikh sistem intellektnogo upravlenija. M.: LENAND \\ URSS, 2015.
6. Masina O.N., Druzhinina O.V. On optimal control of dynamical systems described by differential inclusions // Proceedings of the VII International conference on optimization methods and applications «Optimization and application» (OPTIMA-2016) held in Petrovac, Montentgro, September 25 - October 2, 2016. Moscow: Dorodnicyn Computing Centre of FRC CSC RAS, 2016. P. 104-105.
7. Aleksandrov V.V., Boltjanskijj V.G., Lemak S.S., Parusnikov N.A., Tikhomirov V.M. Optimizacija dinamiki upravljaemykh sistem. M.: Izd-vo MGU. 2000.
8. Aleksandrov V.V., Boltjanskijj V.G., Lemak S.S., Parusnikov N.A., Tikhomirov V.M. Optimalnoe upravlenie dvizheniem. M.: Fizmatlit. 2005.
9. Galeev EH.M., Zelikin M.I., Konjagin S.V., Magaril-Iljaev G.G., Osmolovskijj N.P., Protasov V.JU., Tikhomirov V.M., Fursikov A.V. Optimalnoe upravlenie. M.: MCNMO. 2008.
10. Filippov A.F. O nekotorykh voprosakh teorii optimalnogo regulirovanija // Vestnik MGU. 1959. № 2. S. 25-32.
11. Blagodatskikh V.I. Dostatochnoe uslovie optimalnosti // Differencialnye uravnenija. 1973. T. 9. № 3. S. 416-422.
12. Blagodatskikh V.I., Filippov A.F. Differencialnye vkljuchenija i optimalnoe upravlenie // Tr. MIAN SSSR. 1985. T. 169. S. 194-252.
13. Pontrjagin L.S. Matematicheskaja teorija optimalnykh processov i differencialnye igry // Tr. MIAN SSSR. 1985. T. 169. C. 119-158.
14. Filippov A.F. Klassicheskie reshenija differencialnykh uravnenijj s mnogoznachnojj pravojj chastju // Vestnik MGU. Ser. Mat., mekh. 1967. № 3. S. 16-26.
15. Filippov A.F. Differencialnye uravnenija s razryvnojj pravojj chastju. M.: Nauka. 1985.
16. Tolstonogov A.A. Differencialnye vkljuchenija v banakhovom prostranstve. Novosibirsk: Nauka. 1986.
17. Panasjuk A.I. O dinamike mnozhestv, opredeljaemykh differenci­alnymi vkljuchenijami // Sibirskijj matematicheskijj zhurnal. 1986. T. 27. № 5. S. 155-165.
18. Antosiewicz H.A., Cellina A. Continues selections and differential relations // J. Diff. Eq. 1975. V. 19. P. 386-398.
19. Sean S.W. Existents of solutions and asymptotic equilibrium of multivalued differential systems // J. Math. Anal. and Appl. 1987. V. 89. P. 648-663.
20. Bajjdosov V.A. Nechetkie differencialnye vkljuchenija // Prikladnaja matematika i mekhanika. 1990. T. 54. Vyp. 1. S. 12-17.
21. Masina O.N. O sushhestvovanii reshenijj differencialnykh vkljuchenijj // Differencialnye uravnenija. 2008. T. 44. № 6. S. 845-847.
22. Druzhinina O.V., Masina O.N. Metody issledovanija ustojjchivosti i upravljaemosti nechetkikh i stokhasticheskikh dinamicheskikh sistem. M.: VC RAN. 2009.
23. Balabaeva N.P. Ustojjchivost differencialnykh vkljuchenijj po medlennym peremennym // Nauka i mir. 2015. T. 1. № 3. S. 19-21.
24. Balabaeva N.P. Issledovanie ustojjchivosti differencialnykh vkljuchenijj metodom usrednenija: Diss. - kand. fiz.-mat. nauk. Samara. 2005.
25. Klimov V.S. Teoremy ob ustojjchivosti po pervomu priblizheniju dlja differencialnykh vkljuchenijj // Matem. zametki. 2004. T. 76. № 4. S. 517-530.
26. SHestakov A.A. Obobshhennyjj prjamojj metod Ljapunova dlja sistem s raspredelennymi parametrami. M.: URSS, 2007.
27. Merenkov JU.N. Ustojjchivopodobnye svojjstva differencialnykh vkljuchenijj, nechetkikh i stokhasticheskikh differencialnykh uravnenijj. Monografija. M.: Izd-vo RUDN. 2000.
28. Bacciotti A. Stability and stabilization of discontinuous systems and nonsmooth Lyapynov functions // Rapporto Interno. 1998. № 27. P. 1-21.
29. SHestakov A.A., Stepanov A.N. Indeksnye i divergentnye priznaki ustojjchivosti osobojj tochki avtonomnojj sistemy differencialnykh uravnenijj // Differencialnye uravnenija. 1979. T. 18. № 4. S. 650-661.
30. Druzhinina O.V. Indeksno-divergentnyjj metod issledovanija ustojjchivosti nelinejjnykh dinamicheskikh sistem. M.: VC RAN, 2007.
31. KHajjkin S. Nejjronnye seti: polnyjj kurs. M.: Viljams, 2006.