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Russia Perm
Year
2019
Volume
29
Issue
1
Pages
40-51
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Section Mathematics
Title The structure of the Cauchy operator to a linear continuous-discrete functional differential system with aftereffect and some properties of its components
Author(-s) Maksimov V.P.a
Affiliations Perm State National Research Universitya
Abstract In this paper, a class of linear functional differential systems with aftereffect, continuous and discrete times, and impulses (impulse hybrid systems) is considered. The focus of attention is on the structure of the Cauchy operator to the hybrid system under consideration and the representation of their components. Those allow one to give the representation of all trajectories of the hybrid system and to formulate conditions of the solvability for control problems in various classes of controls, to obtain estimates of the attainability sets under constrained control, and to study general linear boundary value problems for the solvability. A detailed description of all components to the Cauchy operator is given and their properties are studied. For the components with continuous time, some conditions of the continuity with respect to the second argument are obtained which is related to deciding on a class of controls. The main results are based on constructions of the Cauchy matrices to systems with continuous time and difference systems.
Keywords linear systems with delay, functional differential systems with continuous and discrete times, representation of solutions, Cauchy operator
UDC 517.929
MSC 34K10, 34K30, 34K35, 91B74
DOI 10.20537/vm190104
Received 1 February 2019
Language English
Citation Maksimov V.P. The structure of the Cauchy operator to a linear continuous-discrete functional differential system with aftereffect and some properties of its components, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 1, pp. 40-51.
References
  1. Agranovich G. Some problems of discrete/continuous systems stabilization, Funct. Differ. Equ., 2003, vol. 10, no. 1-2, pp. 5-17. https://zbmath.org/?q=an:1175.93185
  2. Agranovich G. Observability criteria of linear discrete-continuous system, Funct. Differ. Equ., 2009, vol. 16, no. 1, pp. 35-51. https://zbmath.org/?q=an:1172.93010
  3. Andrianov D.L. Boundary value problems and control problems for linear difference systems with aftereffect, Russian Mathematics, 1993, vol. 37, no. 5, pp. 1-12. https://zbmath.org/?q=an:0836.34087
  4. Anokhin A.V. On linear impulse systems for functional differential equations, Soviet. Math. Dokl., 1986, vol. 33, pp. 220-223. https://zbmath.org/?q=an:0615.34064
  5. Ashordia M., Chania M., Kucia M. On the solvability of the periodic problem for systems of linear generalized ordinary differential equations, Mem. Differ. Equ. Math. Phys., 2018, vol. 74, pp. 7-26. https://zbmath.org/?q=an:1398.34036
  6. Azbelev N.V., Maksimov V.P., Rakhmatullina L.F. Vvedenie v teoriyu funktsional'no-differentsial'nykh uravnenii (Introduction to the theory of functional differential equations), Moscow: Nauka, 1991.
  7. Azbelev N.V., Maksimov V.P., Rakhmatullina L.F. Elementy sovremennoi teorii funktsional'no-differentsial'nykh uravnenii. Metody i prilozheniya (Elements of the contemporary theory of functional differential equations. Methods and applications), Moscow-Izhevsk: Institute of Computer Science, 2002.
  8. Azbelev N.V., Maksimov V.P., Rakhmatullina L.F. Introduction to the theory of functional differential equations: methods and applications, New York-Cairo: Hindawi Publishing Corporation, 2007.
  9. Azbelev N.V., Maksimov V.P., Simonov P.M. Theory of functional differential equations and applications, International Journal of Pure and Applied Mathematics, 2011, vol. 69, no. 2, pp. 203-235. https://zbmath.org/?q=an:1228.34001
  10. Chadov A.L., Maksimov V.P. Linear boundary value problems and control problems for a class of functional differential equations with continuous and discrete times, Funct. Differ. Equ., 2012, vol. 19, no. 1-2, pp. 49-62. https://zbmath.org/?q=an:1322.34077
  11. Kurzweil Ja. Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Mathematical Journal, 1957, vol. 7, pp. 418-449. https://zbmath.org/?q=an:0090.30002
  12. Maksimov V.P. The Cauchy formula for a functional differential equation, Differ. Uravn., 1977, vol. 13, no. 4, pp. 601-606 (in Russian). http://mi.mathnet.ru/eng/de3033
  13. Maksimov V.P. Voprosy obshchei teorii funktsional'no-differentsial'nykh uravnenii (Problems of the general theory of functional differential equations), Perm: Perm State University, 2003.
  14. Maksimov V.P. Theory of functional differential equations and some problems in economic dynamics, Proceedings of Conference on Differential and Difference Equations and Applications, New York-Cairo: Hindawi Publishing Corporation, 2006, pp. 757-765.
  15. Maksimov V.P. Some questions of the control theory for functional differential systems, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2015, issue 2 (46), pp. 112-119 (in Russian). http://mi.mathnet.ru/eng/iimi310
  16. Maksimov V.P. Continuous mathematical models, Perm: Perm State University, 2015.
  17. Maksimov V.P. On a linear optimal control problem for linear functional differential systems with continuous and discrete times, Functional Differential Equations: Theory and Applications: Proceedings of Conference Dedicated to the 95th Birthday Anniversary of Professor N.V. Azbelev, Perm National Research Polytechnic University, Perm, 2018, pp. 134-141 (in Russian). https://elibrary.ru/item.asp?id=35107826
  18. Maksimov V.P. On a class of optimal control problems for functional differential systems, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2018, vol. 24, no. 1, pp. 131-142 (in Russian). https://doi.org/10.21538/0134-4889-2018-24-1-131-142
  19. Maksimov V.P. Reliable computing experiment in the study of functional-differential equations: theory and applications, Journal of Mathematical Sciences, 2018, vol. 230, issue 5, pp. 712-716. https://doi.org/10.1007/s10958-018-3775-3
  20. Maksimov V.P. Attainable values of on-target functionals for a functional differential system with impulses, Vestn. Tambov. Univ. Ser. Estestv. Tekh. Nauki, 2018, vol. 23, no. 123, pp. 441-447 (in Russian). https://doi.org/10.20310/1810-0198-2018-23-123-441-447
  21. Maksimov V.P., Chadov A.L. Hybrid models in problems of economic dynamics, Perm University Herald. Economy, 2011, no. 2, pp. 13-23 (in Russian). https://elibrary.ru/item.asp?id=17328178
  22. Maksimov V.P., Chadov A.L. A class of controls for a functional-differential continuous-discrete system, Russian Mathematics, 2012, vol. 56, issue 9, pp. 62-65. https://doi.org/10.3103/S1066369X12090083
  23. Maksimov V.P., Rumyantsev A.N. Boundary value problems and problems of pulse control in economic dynamics. Constructive study, Russian Mathematics, 1993, vol. 37, no. 5, pp. 48-62. https://zbmath.org/?q=an:0835.90017
  24. Marchenko V.M. Hybrid discrete-continuous systems with control: II. State space method, Differential Equations, 2015, vol. 51, no. 1, pp. 54-68. https://doi.org/10.1134/S0012266115010061
  25. Marchenko V.M. Controllability and observability of hybrid discrete-continuous systems in the simplest function classes, Differential Equations, 2015, vol. 51, no. 11, pp. 1461-1475. https://doi.org/10.1134/S0012266115110075
  26. Marchenko V.M. Modal control of hybrid differential-difference systems and associated delay systems of neutral type in scales of differential-difference controllers, Differential Equations, 2017, vol. 53, no. 11, pp. 1458-1474. https://doi.org/10.1134/S0012266117110088
  27. Šchwabik S. Generalized ordinary differential equations, Singapore: World Scientific, 1992.
  28. Šchwabik S., Tvrdý M., Veivoda O. Differential and integral equations. Boundary value problems and adjoints, Prague: Academia, 1979.
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