ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2019. Vol. 27

On the Problem of Optimal Stabilization of a System of Linear Loaded Differential Equations

V. R. Barseghyan, T. A. Simonyan, T. V. Barseghyan

The possibilities of modern computing and measuring technologies allow using the most adequate mathematical models for the actual content of controlled dynamic processes. In particular, the mathematical description of many processes of control from various fields of science and technology can be realized with the help of loaded differential equations. In this paper we study the problem of optimal stabilization of one system of linear loaded differential equations. It is assumed that at the loading points the phase-state function of the system has left-side limits. Similar problems arise, for example, when in case of necessity to conduct an observation of a dynamic process, phase states at some moments of time are measured and information continuously is transmitted through a feedback. These problems have important practical and theoretical significance; hence the necessity for their investigations in various settings naturally arises. Taking into account the nature of the influence of the loaded terms on the dynamics of the process, the system of loaded differential equations is represented in the form of stage-by-stage change differential equations. To solve the problem of optimal stabilization of the motion of a stage-by-stage changing system, the problem is divided into two parts, one of which is formulated on a finite time interval, and the second one - on an infinite interval. The problems set up are solved on the basis of the Lyapunov function method. A constructive approach to construct an optimal stabilizing control is proposed.

About the Authors

Vanya Barseghyan, Dr. Sci. (Phys.–Math.), Prof., Leading Scientific Researcher, Institute of Mechanics of NAS of RA, 24 B, Marshal Baghramyan ave., Yerevan, 0019, Armenia; Professor of Mathematics and Mechanics Department, Erevan State University, 1, Alec Manukyan st., Yerevan, 0025, Armenia, e-mail: barseghyan@sci.am

Tamara Simonyan, Cand. Sci. (Phys.–Math.), Assoc. Prof., Mathematics and Mechanics Department, Erevan State University, 1, Alec Manukyan st., Yerevan, 0025, Armenia, e-mail: simtom09@gmail.com

Tigran Barseghyan, Cand. Sci. (Phys.–Math.), Scientific Researcher of Institute of Mechanics of NAS of RA, 24 B, Marshal Baghramyan ave., Yerevan, 0019, Armenia, e-mail: t.barseghyan@mail.ru

For citation

Barseghyan V.R., Simonyan T.A., Barseghyan T.V. On the Problem of Optimal Stabilization of a System of Linear Loaded Differential Equations. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 27, pp. 71-79. (in Russian) https://doi.org/10.26516/1997-7670.2019.27.71

loaded differential equations, differential equations with memory, optimal stabilization problem, Lyapunov function
517.934; 517.977
  1. Albrecht E.G., Shelementyev G.S. Lektsii po teorii stabilizatsii [Lectures on the theory of stabilization]. Sverdlovsk, 1972, 274 p. (in Russian).
  2. Andreev A.S., Rumyantsev V.V. On stabilization of motion of a non-stationary controlled system. Automation and Remote Control, 2007, vol. 68, no. 8, pp. 1309–1321. https://doi.org/10.1134/S0005117907080036
  3. Bakirova E.A., Kadirbayeva Zh.M. O razreshimosti lineynoy mnogotochechnoy krayevoy zadachi dlya nagruzhennykh differentsial’nykh uravneniy [On a Solvability of Linear Multipoint Boundary Value Problem for the Loaded Differential Equations]. Izvestiya HAH PK. Ser. fiz.-mat., 2016, vol. 5, no. 309, pp. 168-175. (in Russian).
  4. Barseghyan V.R. The control problem for a system of linear loaded differential equations with nonseparated multi-point intermediate conditions. Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya Matematika [The Bulletin of Irkutsk State University. Series Mathematics], 2017, vol. 21, pp. 19-32 (in Russian). https://doi.org/10.26516/1997-7670.2017.21.19
  5. Barseghyan V.R, Shaninyan S.G., Barseghyan T.V. Ob odnoy zadache optimal’noy stabilizatsii lineynymi sostavnymi sistemami [About one problem of optimal stabilization of linear compound systems]. Proceedings of NAS RA, Mechanics, 2014, vol. 67, no. 4, pp. 40-52 (in Russian).
  6. Barseghyan V.R. Upravleniye sostavnykh dinamicheskikh sistem i sistem s mnogotochechnymi promezhutochnymi usloviyami [Control of Compound Dynamic Systems and of Systems with Multipoint Intermediate Conditions]. Moscow, Nauka Publ., 2016, 230 p. (in Russian).
  7. Dzhenaliev M.T., Ramazanov M.I. Nagruzhennyye uravneniya kak vozmushcheniya differentsial’nykh uravneniy [Loaded Equations as a Perturbation of Differential Equations]. Almaty, 2010, 334 p. (in Russian).
  8. Kozhanov A.I. Nonlinear loaded equations and inverse problems. Computational Mathematics and Mathematical Physics, 2004, vol. 44, no. 4, pp. 657-678.
  9. Krasovskiy N.N. Problemy stabilizatsii upravlyayemykh dvizheniy [Problems of stabilization of controlled motions]. In the book ”‘Teoriya ustoychivosti dvizheniya”’ [Theory of stability of motion], Moscow, Nauka Publ., 1966, pp. 475-514. (in Russian).
  10. Merkin D.R. Vvedeniye v teoriyu ustoychivosti dvizheniya [Introduction to the theory of stability of motion]. Moscow, Nauka Publ., 1987, 304 p. (in Russian).
  11. Nakhushev A.M. Nagruzhennyye uravneniya i ikh primeneniye [Loaded Equations and their Applications]. Moscow, Nauka Publ., 2012, 232 p. (in Russian).
  12. Shchennikova E.V., Druzhinina O.V., Mulkijan A.S. Ob optimal’noy stabilizatsii mnogosvyaznykh upravlyayemykh sistem [On the optimal stabilization of multiple connected control systems]. Proceedings of the Institute of System Analysis of the Russian Academy of Sciences. Dynamics of nonhomogeneous systems. 2010, vol. 53(3), pp. 99-102. (in Russian).
  13. Barseghyan V.R., Barseghyan T.V. Control problem for a system of linear loaded differential equations. Journal of Physics: Conference Series, 2018, vol. 991, no. 1, art. numb. 012010. https://doi.org/10.1088/1742-6596/991/1/012010
  14. Barseghyan V.R. Control of Stage by Stage Changing Linear Dynamic Systems. Yugoslav Journal of Operations Resarch, 2012, vol. 22, no. 1, pp. 31-39. https://doi.org/10.2298/YJOR111019002B

Full text (russian)