On asymptotically optimal controls in problems with fast and slow variables

I. A. Boitsova

doi:10.15330/ms.44.1.45-55

The problem of optimal control for a system of ordinary differential equations with fast and slow variables and the terminal criterion of quality is studied. It is proved that for any admissible control of the initial system or averaged system there exists the corresponding trajectories of this systems that are approximately equal on the asymptotically large time interval. It is proved that an optimal control of the averaging problem is also an asymptotically optimal control for the two-scale problem.

1. Volosov V.M. Averaging in systems of ordinary differential equations// Uspehi Mat. Nauk. – 1962. – №6. – P. 3–126. (in Russian)

2. Volosov V.M., Morgunov B.I. Averaging method in the teory of nonlinear oscilation systems. – Moscow: Moscow University, 1971. (in Russian)

3. Mitropolsky Yu.A. Averaging method in nonlinear machanics. – Kiev: Naukova Dumka, 1971. (in Russian)

4. Mitropolsky Yu.A., Homa G.P. Mathematical background of nonlinear mechanics asymptotic methods. – Kiev: Naukova Dumka, 1983. (in Russian)

5. Mitropolsky Yu.A., Samoilenko A.M. On the question of asymptotic expansion in nonlinear mechanics// Ukr. Math. Journal. – 1979. – №1. – P. 42–53. (in Russian)

6. Filatov A.N. Averaging methods for differential and integrodifferential equations. – Tashkent: FAN, 1971. (in Russian)

7. Filatov A.N. Asymptotic methods in theory of differential and integrodifferential equations. – Tashkent: FAN, 1974. (in Russian)

8. Bainov D.D., Konstantinov M.M., Milusheva S.D. A boundary-value problem for systems of differential equations with fast and slow variables// Ukr. Math. Journal. – 1980. – V.32, №3. – P. 301–313. (in Russian)

9. Khapaev М.М. Averaging in stability theory – Moscow: Nauka, 1986. (in Russian)

10. Khapaev М.М. Asymptotic methods and stability in nonlinear oscillation theory. – Moscow: Vyshaya Shkola, 1988. (in Russian)

11. Khapaev М.М. Averaging and stability. – Moscow: Znanie, 1991. (in Russian)

12. Moiseev N.N. Asymptotic methods in nonlinear mechanics. – Moscow: Nauka, 1981. (in Russian)

13. Moiseev N.N. Some fragments of optimal systems. – Moscow: Nauka, 1975. (in Russian)

14. Chernousko F.L., Akulenko L.D., Sokolov B.N. Control of oscillations. – Moscow: Nauka, 1980. (in Russian)

15. Petryshyn R.I., Petryshyn Y.R. The averaging method of solving boundary-value problems for differential equation systems with slow and fast variables// Nonlinear Oscillations. – 1998. – №1. – P. 51–65. (in Ukranian)

16. Plotnikov V.A. The averaging method in control problems. – Kiev-Odessa: Lybid, 1992. (in Russian)

17. Plotnikov V.A., Benumerova T.V., Boitsova I.A., Kichmarenko O.D. Averaging equations of controlled motion// Automatics, 2000: International conference on control, Lviv, 2000, V.1. – P. 189–193. (in Russian)

18. Plotnikov V.A., Boitsova I.A. Averaging in problems of the optimal control of systems with fast and slow variables// Problems of control and information. – 2000. – №5. – P. 152–156. (in Russian)

19. Samoilenko A.M., Stanzhitsky A.N. On averaging differential equations on an infinite interval// Differential Equations. – 2006. – V.42, №4. – P. 476–482. (in Russian)

20. Dobrodzii T.V. An averaging method for periodic system control problems// Nonlinear Oscillations. – 2010. – V.13, №2. – P. 147–154. (in Ukranian)

21. Stanzhitsky A.N., Dobrodzii T.V. Study of optimal control problems on the half-line by the averaging method// Differential Equations. – 2011. – V.47, №2. – P. 264–277. (in Russian)

22. Boitsova I.A. Averaging in optimal control problem with fast and slow variables and linear control// Bukovynsky Math. Journal. – 2015. – V.3, №1. – P. 7–15. (in Ukranian)

23. Lee E.B., Markus L. Foundations of optimal control theory. – Moscow: Nauka, 1972. (in Russian)

	On asymptotically optimal controls in problems with fast and slow variables(in Ukrainian)
Author	I. A. Boitsova boitsova.irina@mail.ru Odessa I.I.Mechnikov National University
Abstract	The problem of optimal control for a system of ordinary differential equations with fast and slow variables and the terminal criterion of quality is studied. It is proved that for any admissible control of the initial system or averaged system there exists the corresponding trajectories of this systems that are approximately equal on the asymptotically large time interval. It is proved that an optimal control of the averaging problem is also an asymptotically optimal control for the two-scale problem.
Keywords	task of optimal control; system of differential equations with fast and slow variables; method of averaging; asymptotically optimal control
DOI	doi:10.15330/ms.44.1.45-55
Reference	1. Volosov V.M. Averaging in systems of ordinary differential equations// Uspehi Mat. Nauk. – 1962. – №6. – P. 3–126. (in Russian) 2. Volosov V.M., Morgunov B.I. Averaging method in the teory of nonlinear oscilation systems. – Moscow: Moscow University, 1971. (in Russian) 3. Mitropolsky Yu.A. Averaging method in nonlinear machanics. – Kiev: Naukova Dumka, 1971. (in Russian) 4. Mitropolsky Yu.A., Homa G.P. Mathematical background of nonlinear mechanics asymptotic methods. – Kiev: Naukova Dumka, 1983. (in Russian) 5. Mitropolsky Yu.A., Samoilenko A.M. On the question of asymptotic expansion in nonlinear mechanics// Ukr. Math. Journal. – 1979. – №1. – P. 42–53. (in Russian) 6. Filatov A.N. Averaging methods for differential and integrodifferential equations. – Tashkent: FAN, 1971. (in Russian) 7. Filatov A.N. Asymptotic methods in theory of differential and integrodifferential equations. – Tashkent: FAN, 1974. (in Russian) 8. Bainov D.D., Konstantinov M.M., Milusheva S.D. A boundary-value problem for systems of differential equations with fast and slow variables// Ukr. Math. Journal. – 1980. – V.32, №3. – P. 301–313. (in Russian) 9. Khapaev М.М. Averaging in stability theory – Moscow: Nauka, 1986. (in Russian) 10. Khapaev М.М. Asymptotic methods and stability in nonlinear oscillation theory. – Moscow: Vyshaya Shkola, 1988. (in Russian) 11. Khapaev М.М. Averaging and stability. – Moscow: Znanie, 1991. (in Russian) 12. Moiseev N.N. Asymptotic methods in nonlinear mechanics. – Moscow: Nauka, 1981. (in Russian) 13. Moiseev N.N. Some fragments of optimal systems. – Moscow: Nauka, 1975. (in Russian) 14. Chernousko F.L., Akulenko L.D., Sokolov B.N. Control of oscillations. – Moscow: Nauka, 1980. (in Russian) 15. Petryshyn R.I., Petryshyn Y.R. The averaging method of solving boundary-value problems for differential equation systems with slow and fast variables// Nonlinear Oscillations. – 1998. – №1. – P. 51–65. (in Ukranian) 16. Plotnikov V.A. The averaging method in control problems. – Kiev-Odessa: Lybid, 1992. (in Russian) 17. Plotnikov V.A., Benumerova T.V., Boitsova I.A., Kichmarenko O.D. Averaging equations of controlled motion// Automatics, 2000: International conference on control, Lviv, 2000, V.1. – P. 189–193. (in Russian) 18. Plotnikov V.A., Boitsova I.A. Averaging in problems of the optimal control of systems with fast and slow variables// Problems of control and information. – 2000. – №5. – P. 152–156. (in Russian) 19. Samoilenko A.M., Stanzhitsky A.N. On averaging differential equations on an infinite interval// Differential Equations. – 2006. – V.42, №4. – P. 476–482. (in Russian) 20. Dobrodzii T.V. An averaging method for periodic system control problems// Nonlinear Oscillations. – 2010. – V.13, №2. – P. 147–154. (in Ukranian) 21. Stanzhitsky A.N., Dobrodzii T.V. Study of optimal control problems on the half-line by the averaging method// Differential Equations. – 2011. – V.47, №2. – P. 264–277. (in Russian) 22. Boitsova I.A. Averaging in optimal control problem with fast and slow variables and linear control// Bukovynsky Math. Journal. – 2015. – V.3, №1. – P. 7–15. (in Ukranian) 23. Lee E.B., Markus L. Foundations of optimal control theory. – Moscow: Nauka, 1972. (in Russian)
Pages	45-55
Volume	44
Issue	1
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On asymptotically optimal controls in problems with fast and slow variables(in Ukrainian)