The possibilities of modern computational and measurement techniques allow using the most adequate mathematical models of control of the considered dynamic processes depending on their adequate practical purpose. A mathematical description of various dynamic processes of control in which the future flow of processes depends not only on the present but also is significanlty determined by the history of the process, is performed by using ordinary differential equations with memory of various types, also called equations with aftereffect or loaded differential equations. In this work the problem of control and optimal control of a system of linear loaded differential equations is considered, for which, along with classical boundary (initial and terminal) conditions, nonseparated multipoint intermediate conditions are given. It is assumed that at the loading points the phase-state function of the system has left-side limits and some non-separated multipoint conditions are satisfied. Similar problems arise, for example, when, during the observation of a dynamic process, phase states are measured at some moments of time and the information is continuously transmitted by feedback. These problems have important practical and theoretical value, hence the need for their investigation in various formulations arises naturally. In this work necessary and sufficient conditions for complete controllability for the considered system of linear loaded differential equations is formulated. A constructive approach for the solution of control problem is given and conditions for the existence of program control and motion are formulated. An explicit form of the control action for the control problem is constructed and a method for solving the optimal control problem is proposed. An analitical form of the control action for the control problem is constructed, as well as an approach for solving the optimal control problem is proposed.

MSC

93C15

DOI

https://doi.org/10.26516/1997-7670.2017.21.19

1. Abdullaev V.M. Reshenie differentsial’nykh uravneniy s nerazdelennymi mnogotochechnymi i integral’nymi usloviyami [Solution of Differential Equations with Nonseparated Multipoint and Integral Conditions]. Sibirskii ZhurnalIndustrial’noi Matematiki, 2012, vol. 15, no 3 (51), pp. 3-15.

2. Aschepkov L.T. Optimal’noe upravlenie sistemoy s promezhutochnymi usloviyami [Optimal Control of the System with Intermediate Conditions]. PMM, 1981, vol.45, no 2, pp. 215-222.

3. Bakirova E.A., Kadirbayeva Zh.M. O razreshimosti lineynoy mnogotochechnoy kraevoy zadachi dlya nagruzhennykh differentsial’nykh uravneniy [On a Solvability of Linear Multipoint Boundary Value Problem for the Loaded DifferentialEquations]. Izvestiya HAH PK. Ser. fiz.-mat., 2016, vol. 5, no 309, pp. 168-175.

4. Barseghyan V.R. Upravlenie sostavnykh dinamicheskikh sistem i sistem s mnogotochechnymi promezhutochnymi usloviyami [Control of Compound Dynamic Systems and of Systems with Multipoint Intermediate Conditions]. Moscow, Nauka, 2016. 230 p.

5. Barseghyan V.R., Barseghyan T.V. On an Approach to the Problems of Control of Dynamic System with Nonseparated Multipoint Intermediate Conditions. [Automation and Remote Control], 2015, vol. 76, no 4, pp. 549-559.https://doi.org/10.1134/S0005117915040013

6. 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.

7. Barseghyan V.R. Upravlenie poetapno menyayushchimisya lineynymi dinamicheskimi sistemami s ogranicheniyami na znacheniya chastey koordinat fazovogo vektora v promezhutochnye momenty vremeni [Control of Stage by StageChanging Linear Dynamic Systems with Constraints on the Values of Parts of the Coordinates of the Phase Vector in the Intermediate Moments of Time]. Dinamika neodnorodnykh sistem. Trudy ISA RAN, 2010, vol. 53(2), no 14, pp. 7-18.

8. Dzhenaliev M.T., Ramazanov M.I. Nagruzhennye uravneniya kak vozmushcheniya differentsial’nykh uravneniy [Loaded Equations as a Perturbation of Differential Equations]. Almaty, 2010. 334 p.

9. Dzhumabaev D.S., Imanchiev A.E. Korrektnaya razreshimost’ lineynoy mnogotochechnoy kraevoy zadachi [Correct Solvability of Linear Miltipoint Boundary Value Problem]. Matematicheskiy zhurnal, 2005, vol. 5, no 1, pp. 30-38.

10. Dykhta V.A., Samsonyuk O.N. Printsip maksimuma dlya gladkikh zadach optimal’nogo impul’snogo upravleniya s mnogotochechnymi fazoogranicheniyami [The Maximum Principle for Smooth Problems of Optimal Impulse Controlwith Multipoint Phase Constraints]. Zhurnal Vichislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, vol.49, no 6, pp. 981-997.

11. Zubov V.I. Lektsii po teorii upravleniya [Lectures on the Theory of Control]. Moscow, Nauka, 1975. 496 p.

12. Kozhanov A.I. Nelineynye nagruzhennye uravneniya i obratnye zadachi [Nonlinear Loaded Equations and Inverse Problems]. Zh. Vychisl. Mat. Mat. Fiz., 2004, vol. 44, no 4, pp. 694-716.

13. Konyaev Yu.A. Konstruktivnyemetody issledovaniya mnogotochechnykh kraevykh zadach [Constructive Methods for StudyingMultipoint Boundary Value Problems]. Izv. Vuzov. Matem., 1992, no 2, pp. 57-61.

14. Krasovsky N.N. Teoriya upravleniya dvizheniem [The Theory of Motion Control]. Moscow, Nauka, 1968. 476 p.

15. Mironov V.I., Mironov Yu.V. Energeticheski optimal’noe upravlenie v lineynykh mnogotochechnykh zadachakh o vstreche dvizheniy [The Power Optimum Control in Linear Multipointed Problems on Meeting Points Motions]. Trudy SPIIRAN,2007, no 5, pp. 322-328.

16. Nakhushev A.M. Nagruzhennye uravneniya i ikh primenenie [Loaded Equations and their Applications]. Moscow, Nauka, 2012. 232 p.

17. Samoilenko A.M., Laptinsky V.N., Kenzhebaev K.K. Konstruktivnye metody issledovaniya periodicheskikh i mnogotochechnykh kraevykh zadach [Constructive Methods of Investigation of Periodic and Multipoint Boundary Problems]. Kiev, IM NAN Ukr., 1999. 220 p.

18. Emel’yanov S.V. (ed.) Teoriya sistem s peremennoy strukturoy[Theory of Variable Structure Systems]. Moscow, Nauka, 1970. 592 p.