TOWARDS THE THEORY OF LINEAR DYNAMIC NONANTAGONISTIC GAMES 

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 (Orenburg region, Orsk, 4 Orskoe road), pasikov_fmf@mail.ru 

517.977 

The article studies the problems of the theory of dynamic games of several persons with non-zero sum, when the value of the game is the system of functionals of distance type. The peculiarity of the work lies in the fact that to describe the evolution of objects there may be used three cases of linear systems of Volterra type: integro- differential system of equations with managing impacts outside of the integral, integro-differential system of equations with control actions under the sign of the interval and the system of integral equations. Solution of the problem lies in the construction of equilibrium, the Nash equilibrium, the set of optimal strategies for specified
types of dynamical systems and the selected features. The problem is solved by constructing some modification of the well known extreme construction of the academician N. N. Krasovskiy, which is based on a new definition of the position of the game that uses a full memory of the control inputs, which significantly complicates the entire study. The corresponding theorems have been proven. 

Volterra integral differential equation, Volterre’s integral equation, control action, measurable function, trajectory, game position, optimal strategy. 

1. Krasovskiy N. N. Igrovye zadachi o vstreche dvizheniy [Game problem on motions meeting]. Moscow: Nauka, 1970, 420 p.
2. Subbotin A. I., Chentsov A. G. Optimizatsiya garantii v zadachakh upravleniya [Assurance optimization in control problems]. Moscow: Nauka, 1981, 288 p.
3. Gorokhovik V. V., Kirillova F. M. Upravlyaemye sistemy: sb. tr. – Vyp. 10 [System control: collected papers – Issue 10]. Novosibirsk, 1971, pp. 3–9.
4. Gnedenko B. V. Kurs teorii veroyatnostey [Probability theory course]. Moscow: Nauka, 2005, 448 p.
5. Pasikov V. L. Izvestiya RAEN. Differentsial'nye uravneniya [Bulletin of the Russian Academy of Natural Sciences. Differential equations]. 2008, no. 13, pp. 95–101.
6. Pasikov V. L. Izvestiya vysshikh uchebnykh zavedeniy. Povolzh-skiy region. Fizikomatematicheskie nauki [University proceedings. Volga region. Physics and mathematics sciences]. 2011, no. 2, pp. 58–70.
7. Pasikov V. L. Izvestiya vysshikh uchebnykh zavedeniy. Povolzh-skiy region. Fizikomatematicheskie nauki [University proceedings. Volga region. Physics and mathematics sciences]. 2012, no. 2.
8. Pasikov V. L. Differentsial'nye uravneniya [Differential equations]. 1986, V. XXII, no. 5, pp. 907–909.
9. Pasikov V. L. Igrovye zadachi dlya sistem integral'nykh uravneniy Vol'terra [Gamy problems for Volterra integral equations system]. Ryazan: Ryazanskiy ordena «Znak pocheta» gospedinstitut, 1983, 42 p.