APPROACH OF SINGLE-TYPE OBJECTS, EVOLUTION OF WHICH IS DESCRIBED BY VOLTERRA SYSTEMS

Pasikov Vladimir Leonidovich, Candidate of physical and mathematical sciences, associate professor, sub-department of natural and mathematical disciplines, Orsk branch of Orenburg State Institute of Management (4 Orskoe highway, Orsk, Orenburg region, Russia), pasikov_fmf@mail.ru

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Background. The paper discusses some problems of optimal control, namely, the theory of dynamic games when the game dynamics is described by linear integral and integrodifferential vector Volterra equations. The aim of the article is to solve problems of optimization of distance-type functionals.
Materials and methods. To solve these problems, the author built a modification of the famous extreme construction of academician N. N. Krasovskiy developed for ordinary differential systems. The centerpiece of this modification is a new definition of the game position for which it is necessary to calculate the total memory to man-age stress that greatly complicates the entire study compared with the case of ordi-nary differential systems.
Results and conclusions. The paper presents significant new results that comple-ment and extend the general theory of dynamic games. They consist in the spread of classical methods of academician N. N. Krasovskiy on more complex objects – Volterra dynamic systems. Thus, the author has proved the possibility of extending the field of application of these methods.

Volterra integral equation, Volterra integrodifferential equation, control action, optimal strategy, measurable function, game position.

1. Tsalyuk Z. B. Matematicheskiy analiz. Itogi nauki i tekhniki [Mathematical analysis. Results of science and technology]. Moscow: VINITI, 1977, pp. 131–198.
2. Lando Yu. K. Elementy matematicheskoy teorii upravleniya dvizheniem [Elements of mathematical theory of movement control]. Moscow: Prosveshchenie, 1984, 88 p.
3. Vinokurov V. R. Izvestiya vuzov. Matematika [University proceedings. Mathematics]. 1969, no. 6, pp. 24–34.
4. Krasovskiy N. N. Igrovye zadachi o vstreche dvizheniy [Game problems on encounter of movements]. Moscow: Nauka, 1970, 420 p.
5. Pasikov V. L. Differentsial'nye uravneniya [Differential equations]. 1986, vol. XXII, no. 5, pp. 907–909.
6. Filippov A. F. Differentsial'nye uravneniya s razryvnoy pravoy chast'yu [Differential equations with discontinuous right part]. Moscow: Nauka, 1985, 224 p.
7. Pasikov V. L. Vestnik Yuzhno-ural'skogo gosudarstvennogo universiteta. Ser. Mate-matika. Mekhanika. Fizika [Bulletin of South-Ural State University. Series: Mathemat-ics. Mechanics]. 2012, vol. 7,no.34(293),pp.33–42.
8. Pasikov V. L. Izvestiya vysshikh uchebnykh zavedeniy Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathemati-cal sciences]. 2011, no. 2 (18), pp. 58–70.
9. Pasikov V. L. Vestnik Yuzhno-ural'skogo gosudarstvennogo universiteta. Ser. Mate-matika. Mekhanika. Fizika [Bulletin of South-Ural State University. Series: Mathemat-ics. Mechanics. Physics]. 2013, vol. 5,no.1,pp.55–62.
10. Pasikov V. L. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University proceedings. Volga region. Physical and mathemati-cal sciences]. 2012, no. 2 (22), pp. 50–58.