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

List of issues > Series «Mathematics». 2020. Vol. 31

On Necessary Optimality Conditions for Discrete Control Systems

M. J. Mardanov, T. K. Melikov

In this article, under weakened assumptions, we study high-order necessary optimality conditions for discrete optimal control problems with the free right end of the trajectory. Here, we first use the concept of the relative interior of a set in the broad sense, and then the combination of linear (i.e. uniformly small) and needle variation of the admissible control. As a result, a new formula for the increment of the quality functional with the members of zeroth, first and second order of smallness is obtained. This formula serves as a source of the well-known zeroth order necessary optimality condition, if the admissible control has no linear variation, or the well-known first and second order necessary optimality conditions, if the increment of the quality functional of order zero is vanished on a certain subset of the domain of admissible controls. Following the obtained formula of the increment of the quality functional, the concepts of zeroth, first and second variations of the quality functional are introduced in a more general form, from which, in particular, the well-known variations of the quality functional follow. Based on the obtained formulae for the variations of the quality functional, using the needle variation of the admissible control, more constructive the zeroth, first and second order necessary optimality conditions with broad applications area are obtained.

About the Authors

Misir Mardanov, Dr. Sci. (Phys.–Math.), Prof., Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, 9, ul. B. Vagabzade, Baku, AZ1141, Azerbaijan Republic, тел.: (+99450)5393924, e-mail: misirmardanov@yahoo.com

Telman Melikov, Dr. Sci. (Phys.–Math.), Prof., Institute of Mathematics and Mechanics and Institute of Control System, Azerbaijan National Academy of Sciences, 9, ul. B. Vagabzade, Baku, AZ1141, Azerbaijan Republic, тел.: (+99450) 5393924, e-mail: t.melik@rambler.ru

For citation

Mardanov M.J., Melikov T.K. On the Necessary Conditions of Optimality in Discrete Control Systems. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 31, pp. 49-61. (in Russian) https://doi.org/10.26516/1997-7670.2020.31.49

discrete control systems, optimal control, necessary conditions, variations of cost function

1. Ashchepkov L.T. Optimalnoe upravlenie s razryvnymi sistemami [Optimal Control of Discontinuous Systems]. Novosibirsk, Nauka Publ., 1987, 226 p. (in Russian)

2. Boltyanskii V.G. Optimalnoe upravlenie diskretnymi sistemami [Optimal Control of Discrete Systems]. Moscow, Nauka Publ., 1978, 446 p. (in Russian)

3. Butkovskii A.G. O neobkhodimykh i dostatochnykh uslaviyakh optimalnosti dlya impulsnikh sistem upravleniya [On necessary and sufficient optimality conditions for impulse control systems]. Autom. Remote Control, 1963, vol. 24, no. 8, pp. 1056-1064. (in Russian)

4. Gabasov R.F. Ob optimalnosti osobykh upravleniy [The optimality of singular controls]. Differential Equations, 1968, vol. 4, pp. 1000-1011. (in Russian)

5. Gabasov R.F., Kirillova F.M. K theorii neobxodimykh usloviy optimalnosti dlya diskretnykh sistemy [On the theory of necessary conditions for optimality for discrete systems]. Autom. Remote Control., 1969, vol. 12 , no. 30, pp. 1921-1928. (in Russian)

6. Gabasov R.F., Kirillova F.M., Kachestvennaya teoriya optimalnykh protsessov [Qualitative Theory of Optimal Processes]. Moscow, Nauka Publ., 1971, 508 p. (in Russian)

7. Krasovsky N.N. Ob odnoy zadache optimalnogo regulirovaniya [On a problem of optimal regulation.] Applied mathematics and mechanics, 1957, vol. 21, no. 5, pp. 670-677. (in Russian)

8. Mardanov M.J., Melikov T.K. Novyi diskretnyi analog printsipa maksimuma Pontryagina [A New Discrete Analogue of Pontryagin’s Maximum Principle.] Doklady Mathematics, 2018, vol. 98, no. 3, pp. 549–551. https://doi.org/10.1134/S1064562418070049

9. Mardanov M.J., Melikov T.K . Malik S.T. K theorii optimalnykh protsessov v diskretnykh sistemakh [On the Theory of Optimal Processes in Discrete Systems] Math. Notes, 2019, vol. 106, no. 3, pp. 390–401.

10. Mordukhovich B.S. Metody approksimatsii i upravleniya [Approximation methods in optimization and control problems.] Moscow, Nauka Publ., 1988. 360 p. (in Russian)

11. Propoi A.I. Elementy teorii optimalnykh diskternykh protsessov [Elements of the Theory of Optimal Discrete Processes.] Moscow, Nauka Publ., 1973, 256 p. (in Russian)

12. Rozonoer L.I. Printsip maksimuma L.S. Pontryagina v teorii optimalynykh sistem [Pontryagin’s maximum principle in the theory of optimal systems]. Autom. Remote Control, 1959, vol. 20, no. 12, pp. 1561-1578. (in Russian)

13. Fan Liang-Tseng, Wang Chu-Sen. Diskretnyi printsip maksimuma [The discrete maximum principle]. Moscow, Mir Publ., 1967, 180 p.

14. Jordan B.K., Polak E. Theory of class of discrete optimal control system. J. Electr. and Control, 1964, vol. 17, no. 6, pp. 697-711. https://doi.org/10.1080/00207216408937740

15. Mardanov M.J., Melikov T.K, Malik S.T., Malikov K. First- and second-order necessary conditions with respect to components for discrete optimal control problems. J. Comput. Appl. Math., 2020, vol. 364, 15 January, 112342. https://doi.org/10.1016/j.cam.2019.112342

16. Mardanov M.J., Melikov T.K. A method for studying the optimality of controls in discrete systems. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 2014, vol. 40, no 2, pp. 5-13.

17. Mardanov M.J., Melikov T.K. On strengthening of optimality conditions in discrete control systems, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 2018, vol. 44, no. 1, pp. 135-154.

18. Toan N.T., Thuy L.Q. Second-order necessary optimality conditions for a discrete optimal control problem with mixed constrains. Journal of Global Optimization, 2015, vol. 64, no. 3, pp. 533-562. https://doi.org/10.1007/s10898-015-0333-0

Full text (russian)