The control problem for a system of linear loaded differential equations with nonseparated multi-point intermediate conditions

V. R. Barseghyan

The Bulletin of Irkutsk State University. Series Mathematics, Tome 21 (2017), pp. 19-32

Résumé

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 nonseparated 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.

Détail

Keywords: loaded differential equations, differential equations with memory, multipoint intermediate conditions, control problem, complete controllability.

Bibliographie

[1] Abdullaev V. M., “Solution of Differential Equations with Nonseparated Multipoint and Integral Conditions”, Sibirskii Zhurnal Industrial'noi Matematiki, 15:3(51) (2012), 3–15

[2] Aschepkov L. T., “Optimal Control of the System with Intermediate Conditions”, PMM, 45:2 (1981), 215–222 | Zbl

[3] Bakirova E. A., Kadirbayeva Zh. M., “On a Solvability of Linear Multipoint Boundary Value Problem for the Loaded Differential Equations”, Izvestiya HAH PK. Ser. fiz.-mat., 2016, no. 5, 168–175

[4] Barseghyan V. R., Control of Compound Dynamic Systems and of Systems with Multipoint Intermediate Conditions, Nauka, M., 2016, 230 pp.

[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, 76:4 (2015), 549–559 | DOI | MR | Zbl

[6] V. R. Barseghyan, “Control of stage by stage changing linear dynamic systems”, Yugoslav Journal of Operations Resarch, 22:1 (2012), 31–39 | DOI | MR | Zbl

[7] Barseghyan V. R., “Control of Stage by Stage Changing 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, 53(2), no. 14, 2010, 7–18 | Zbl

[8] Dzhenaliev M. T., Ramazanov M. I., Loaded Equations as a Perturbation of Differential Equations, Almaty, 2010, 334 pp.

[9] Dzhumabaev D. S., Imanchiev A. E., “Correct Solvability of Linear Miltipoint Boundary Value Problem”, Matematicheskiy zhurnal, 5:1(15) (2005), 30–38

[10] Dykhta V. A., Samsonyuk O. N., “The Maximum Principle for Smooth Problems of Optimal Impulse Control with Multipoint Phase Constraints”, Zhurnal Vichislitel'noi Matematiki i Matematicheskoi Fiziki, 49:6 (2009), 981–997 | Zbl

[11] Zubov V. I., Lectures on the Theory of Control, Nauka, M., 1975, 496 pp.

[12] Kozhanov A. I., “Nonlinear Loaded Equations and Inverse Problems”, Zh. Vychisl. Mat. Mat. Fiz., 44:4 (2004), 694–716 | Zbl

[13] Konyaev Yu. A., “Constructive Methods for Studying Multipoint Boundary Value Problems”, Izv. Vuzov. Matem., 1992, no. 2, 57–61

[14] Krasovsky N. N., The Theory of Motion Control, Nauka, M., 1968, 476 pp.

[15] Mironov V. I., Mironov Yu. V., “The Power Optimum Control in Linear Multipointed Problems on Meeting Points Motions”, Trudy SPIIRAN, 2007, no. 5, 322–328

[16] Nakhushev A. M., Loaded Equations and their Applications, Nauka, M., 2012, 232 pp.

[17] Samoilenko A. M., Laptinsky V. N., Kenzhebaev K. K., Constructive Methods of Investigation of Periodic and Multipoint Boundary Problems, IM NAN Ukr., Kiev, 1999, 220 pp.

[18] Emel'yanov S. V. (ed.), Theory of Variable Structure Systems, Nauka, M., 1970, 592 pp.