DOI:https://doi.org/10.15407/kvt198.04.040
Cybernetics and Computer Engineering, 2019, 4(198)
CHIKRII G.Ts., DSc. (Phys-Math), Senior Researcher
Leading Researcher of the Economical Cybernetics Department
e-mail: g.chikrii@gmail.com
V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine
40, Acad. Glushkov av., Kyiv, 03187, Ukraine
PRINCIPLE OF TIME STRETCHING IN GAME DYNAMIC PROBLEMS
Introduction. There exists a wide range of mechanical, economical and biological processes evolving in condition of conflict and uncertainty, which can be described by various kind dynamic systems, depending on the process nature. This paper deals with the dynamic games of pursuit, described by a system of general form, encompassing a wide range of the functional-differential systems. The deciding factor in study of dynamic games is availability of information on current state of the process. In real systems information, as a rule, arrives with delay in time. Also, there are a number of problems for which Pontryagin’s condition, reflecting an advantage of the pursuer over the evader in control resources, does not hold. Establishment of close relation between its time-stretching generalization and the effect of variable information delay offers much promise for solving above mentioned problems.
The purpose of the paper is, to establish sufficient conditions for termination of the games on the basis of effect of information delay, for which Pontryagin’s condition does not hold, to specify these conditions for the case of integro-differential dynamics, and to illustrate the obtained result with the model example.
Methods. For investigation of the dynamic game of pursuit we apply the scheme of Pontryagin’s First Direct Method providing bringing of the trajectory of conflict-controlled process to the cylindrical terminal set at a finite moment of time. In so doing, construction of the pursuer’s control is accomplished on the basis of the Filippov-Castaing theorem on measurable choice insures realization of the process of pursuit in the class of stroboscopic strategies by Hajek. To deduce solution of the conflict-controlled integro-differentional system in the form of Cauchy formula, the method of successive approximation is used.
Results. It is shown that the dynamic game of pursuit with separated control blocks of the players and variable delay of information is equivalent to certain perfect information game. Based on this fact, the principle of time stretching is developed to study the games with complete information for which classic Pontryagin’s condition, lying at the heart of all direct methods of pursuit, does not hold. The time-stretching modification of this condition, proposed in the paper, makes it feasible to obtain sufficient conditions for bringing the game trajectory to the terminal set at a finite moment of time. In so doing, the control of pursuer, providing achievement of the game goal, is constructed. These conditions are specified for the integro-diffential game of pursuit. By way of illustration, an example of integro-differential game of pursuit is analyzed in detail. It is found that the time stretching function provides fulfillment of generalized Pontryagin’s condition. Simple relationships between dynamics parameters and control resources of the players are deduced that provide capture of the evader by the pursuer, under arbitrary initial states of the players.
Conclusion. Thus, in the paper an efficient tool is developed for analysis of conflict situation, for example, interception of a mobile target by controlled object in condition of conflict counteraction. Situation is analyzed, when the pursuing object lacks conventional advantage in control resources over the evading counterpart, that is, the classic Pontryagin’s condition does not hold. Suggested approach makes it feasible to realize the process of pursuit with the help of appropriate Krasovskii’ counter-controls.
Keywords: dynamic game, time-variable information delay, Pontryagin’s condition, Aumann’s integral, principle of time stretching, Minkowski’ difference, integro-differential game.
REFERENCES
1 Isaacs R. Differential Games. Moscow, 1967. (in Russian).
2 Pontryagin L.S. Selected Scientific Papers. Vol. 2. Moscow, 1988. (in Russian).
3 Krasovskii N.N. Game Problems on the Encounter of Motions. Moscow, 1970. (in Russian).
4 Berkovitz L.D. Differential games of generalized pursuit and evasion. SIAM Control and Optimization. 1986. Vol. 24. No 3. P. 361-373. https://doi.org/10.1137/0324021
5 Friedman A. Differential Games. New York, 1971.
6 Hajek O. Pursuit Games. New York, 1975.
7 Pshenitchny B.N. -strategies in Differential Games, Topics in Differential Games. New York, London, Amsterdam: North Holland, 1973. P. 45-99.
8 Chikrii A.A. An analytic method in dynamic games. Proceedings of the Steklov Institute of Mathematics. 2010. Vol. 271. P. 69-85. https://doi.org/10.1134/S0081543810040073
9 Dziubenko K.G., Chikrii A.A. An approach problem for a discrete system with random perturbations. Cybernetics an York, Sprid Systems Analysis. 2010. Vol. 46. No.2. P. 271-281. https://doi.org/10.1007/s10559-010-9204-3
10 Dziubenko K.G., Chikrii A.A. On the game problem of searching moving objects for the model of semi-markovian type. Journal of Automation and Information Sciences. 2006. Vol. 38. No.9. P. 1-11.
11 Siouris G. Missile Guidance and Control Systems. New York, 2004. https://doi.org/10.1115/1.1849174
12 Chikrii G.Ts. On a problem of pursuit under variable information time lag on the availability of a state vector. Dokl. Akad. Nauk Ukrainy. 1979. No. 10. P. 855-857 (in Russian)
13 Chikrii G.Ts. An approach to solution of linear differential games with variable information delay. J. Autom. and Inform. Sci., 1995. Vol. 27 (3&4). P. 163-170.
14 Nikolskij M.S. Application of the first direct method in the linear differential games. Izvestia Akad. Nauk SSSR. Vol 10:51-56 (in Russian).
15 Chikrii A.A. Conflict-Controlled Processes. Springer Science & Business Media, 2013.
16 Mezentsev A.V. On some class of differential games. Izvestia AN SSSR. Techn. kib. 1971. No. 6. P.3-7 (in Russian).
17 Zonnevend D. On One Method of Pursuit. Doklady Akademii Nauk SSSR. Vol. 204. P. 1296-1299 (in Russian).
18 Chikrii G.Ts. Using impact of information delay for solution of game problems of pursuit. Dopovidi Natsional’noi Akademii Nauk Ukrainy. Vol 12. P. 107-111.
19 Chikrii G.Ts. One approach to solution of complex game problems for some quasilinear evolutionary systems. Journal of Mathematics, Game Theory and Algebra. 2004. Vol. 14. P. 307-314.
20 Chikrii G.Ts. Using the effect of information delay in differential pursuit games. Cybernetics and Systems Analysis. 2007. Vol. 43. No. 2. P. 233-245. https://doi.org/10.1007/s10559-007-0042-x
21 Chikrii G.Ts. On one problem of approach for damped oscillations. Journal of Automation and Information Sciences. 2009. Vol. 41. No.4. P. 1-9. https://doi.org/10.1615/JAutomatInfScien.v41.i10.10
22 Chikrii G.Ts. Principle of time sretching in evolutionary games of approach. Journal of Automation and Information Sciences. 2016. Vol.48. No. 5. P. 12-26. https://doi.org/10.1615/JAutomatInfScien.v48.i5.20
23 Aumann R.J. Integrals of set-valued functions. J. Math. Anal. Appl. 1965. vol.12. P. 1-12 https://doi.org/10.1016/0022-247X(65)90049-1
24 Filippov A.F. Differential Equations with Discontinuous Right Side. Moscow, 1985 (in Russian).
25 Krasnov M.L., Kiseliov A.I., Makarenko G.I. Integral Equations. Moscow. 1968. (in Russian).
26 Kolmogorov A.N., Fomin S.V. Elements of Theory of Functions and Functional Analysis. Moscow, 1989. (in Russian).
Received 04.09.2019