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

List of issues > Series «Mathematics». 2020. Vol. 32

Second Order Krotov Method for Discrete-Continuous Systems

I. V. Rasina, O. V. Danilenko

In the late 1960s and early 1970s, a new class of problems appeared in the theory of optimal control. It was determined that the structure of a number of systems or processes is not homogeneous and can change over time. Therefore, new mathematical models of heterogeneous structure have been developed.

Research methods for this type of system vary widely, reflecting various scientific schools and thought. One of the proposed options was to develop an approach that retains the traditional assumptions of optimal control theory. Its basis is Krotov's sufficient optimality conditions for discrete systems, formulated in terms of arbitrary sets and mappings.

One of the classes of heterogeneous systems is considered in this paper: discretecontinuous systems (DCSs). DCSs are used for case where all the homogeneous subsystems of the lower level are not only connected by a common functional but also have their own goals.

In this paper a generalization of Krotov's sufficient optimality conditions is applied. The foundational theory is the Krotov method of global improvement, which was originally proposed for discrete processes. The advantage of the proposed method is that its conjugate system of vector-matrix equations is linear; hence, its solution always exists, which allows us to find the desired solution in the optimal control problem for DCSs.

About the Authors

Irina Rasina, Dr. Sci. (Phys.–Math.), Assoc. Prof., Program Systems Institute of RAS, 4a, Petr Pervyi st., 152020, Pereslavl-Zalessky, Russian Federation, tel.: (48535)98028, e-mail: irinarasina@gmail.com

Olga Danilenko, Cand. Sci. (Phys.–Math.), Institute of Control Sciences of RAS, 65. Profsoyuznaya st., 117997, Moscow, Russian Federation, tel.: (495)3349159, e-mail: olga@danilenko.org

For citation

Rasina I. V., Danilenko O. V. Second Order Krotov Method for Discrete-Continuous Systems. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 32, pp. 17-32. https://doi.org/10.26516/1997-7670.2020.32.17

discrete-continuous systems, sufficient optimality conditions, control improvement method
  1. Bortakovskii A.S. Sufficient Conditions of Control Optimality for Determined Logical Dynamic Systems [Dostatochnye uslovija optimal’nosti upravlenija determinirovannymi logiko-dinamicheskimi sistemami]. Informatika. Ser. Avtomatizacija proektirovanija, 1992, vol. 2-3, pp. 72-79. (in Russian)
  2. Bortakovskii A.S. Synthesis of Optimal Control-Systems with a Change of the Models of Motion. Journal of Computer and Systems Sciences International, 2018, vol. 57, no. 4, pp. 543-560. https://doi.org/10.1134/S1064230718040056
  3. Gurman V.I. Theory of Optimum Discrete Processes. Autom. Remote Control, 1973, vol. 34, no. 7, no. 6, pp. 1082-1087. (in Russian)
  4. Gurman V.I. Printsip rasshireniya v zadachakh upravleniya. The expansion principle in control problems. Moscow, Nauka Publ., 1985, 288 p. (in Russian)
  5. Gurman V.I. Abstract Problems of Optimization and Improvement. Program Systems: Theory and Applications. 2011, no. 5(9), pp. 14-20. URL: http://psta.psiras.ru/read/psta2011_5_21-29.pdf. (in Russian)
  6. Gurman V.I., Trushkova E.A. Approximate Methods of Control Processes Optimization. Program Systems: Theory and Applications, 2010, no, 4(4), pp. 85-104. URL: http://psta.psiras.ru/read/psta2010_4_85-104.pdf. (in Russian)
  7. S.V. Emel’yanov. Teorija sistem s peremennoj strukturoj [Theory of Systems with Variable Structure]. Moscow, Nauka Publ., 1970, 592 p. (in Russian) 
  8. Krotov V.F. Sufficient Optimality Conditions for Discrete Controlled Systems [Dostatochnye uslovija optimal’nosti dlja diskretnyh upravljaemyh sistem]. Dokl. Akad. Nauk SSSR. 1967, vol. 172, no. 1, pp. 18-21. (in Russian) 
  9. Krotov V.F., Gurman V.I. Metody i zadachi optimal’nogo upravlenija [Methods and Problems of Optimal Control]. Moscow, Nauka Publ., 1973, 448 p. (in Russian) 
  10. Krotov V.F., Fel’dman I.N. Iterative Method for Solving Optimal Control Problems [Iteracionnyj metod reshenija zadach optimal’nogo upravlenija]. Izvestija AN SSSR. Tehnicheskaja kibernetika, 1983, no. 2, pp. 160-168. (in Russian) 
  11. Miller B.M., Rubinovich E.Ya. Optimizacija dinamicheskih sistem s impul’snymi upravlenijami [Optimization of Dynamic Systems with Impulse Controls]. Moscow, Nauka Publ., 2005, 429 p. (in Russian) 
  12. Rasina I.V. Ierarkhicheskie modeli upravleniya sistemami neodnorodnoy struktury [Hierarchical Models of Control of Heterogeneous Systems]. Moscow, Fizmatlit Publ., 2014, 160 p. (in Russian) 
  13. Rasina I.V. Descrete-Continuous Systems with Intermediate Criteria [Diskretnonepreryvnye sistemy s promezhutochnymi kriteriyami]. Materialy XX Yubileynoy Mezhdunarodnoy konferenysii po vychislitel’noy mekhanike i sovremennym prikladnym programmnym sistemam [Materials of the XX Anniversary International Conference on Computational Mechanics and Modern Applied Software Systems]. Moscow, MAI Publ.,2017, pp. 699-701. (in Russian) 
  14. Lygeros J. Lecture Notes on Hybrid Systems. Cambridge, University of Cambridge, 2003, 84 p. 
  15. Rasina I., Danilenko O. Second-Order Improvement Method for DiscreteContinuous Systems with Intermediate Criteria. IFAC-Papers Online, 2018, vol. 51, no. 32, pp. 184-188. https://doi.org/10.1016/j.ifacol.2018.11.378

Full text (english)