Issue 4 (198), article 3


Cybernetics and Computer Engineering, 2019, 4(198)

CHIKRII G.Ts., DSc. (Phys-Math), Senior Researcher
Leading Researcher of the Economical Cybernetics Department

V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine
40, Acad. Glushkov av., Kyiv, 03187, Ukraine


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.

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Received 04.09.2019