The authors consider a game problem of soft meeting of controlled oscillating systems, i.e., their simultaneous convergence in geometric coordinates and velocities. Applying Pontryagin's first direct method [1] to solve this problem is noted to be impossible since the condition underlying this method is not satisfied. This condition is an instantaneous advantage of the pursuer (the one who strives to achieve this meeting) over the evader (the one who tries to avoid it). In the method, we apply the principle of time dilation, which weakens this condition and makes it possible to terminate the game in a finite time. The paper outlines the problem solution method that employs a certain time dilation function. Also, an algorithm, versions of constructing pursuer's control, and an example of computer implementation of the process of convergence on the plane are provided.

The authors propose a method for solving the game problem where the trajectory of a quasi-linear non-stationary system approaches a cylindrical terminal set varying with time. The case is considered where Pontryagin’s condition (the condition of the first player’s advantage) is not satisfied. A time dilation function is introduced, which postpones the time of game termination and is used to introduce a modified Pontryagin’s condition, which allows making a measurable choice of control. The basic method is the method of resolving functions. Using the technique of set-valued mappings and their selectors, the strategies are generated, which guarantee the problem solution. The process of convergence of the trajectory and the terminal set consists of two sections: active and passive, where the control of the first player is selected using the control of the second player with a certain time delay, which depends on the time dilation function. The scheme of the method is outlined and sufficient conditions for the game termination in a finite time are obtained.

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 modification and the effect of variable information delay offers much promise for solving the above mentioned problems.The purpose of the paper is to deduce, sufficient conditions for termination of the games, for which Pontryagin's condition does not hold, by the use of the effect of information delay, 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 that 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 Cauchy form, 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.The time stretching function, providing fulfillment of the modified Pontryagin's condition is found.Simple relationships between © G.Ts. CHIKRII, 2019 dynamics parameters and control resources of the players are deduced that provide feasibility of 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.The 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.

A conflict-controlled process of the approach of a trajectory to a cylindrical terminal set is studied. The problem statement encompasses a wide range of quasilinear functional-differential systems. We use the technique of set-valued mappings and their selections to derive sufficient conditions for the game termination in a finite time. The methodology used is close to the scheme that involves the time of the first absorption. By way of illustration, quasilinear integro-differential games are examined. For this purpose, their solutions are presented in the form of an analog of the Cauchy formula. The calculations are performed for the case of a system with a simple matrix; the control sets of the players are balls centered at the origin and the terminal set is a linear subspace. Depending on the relations between the initial state of the system and the parameters of the process, sufficient conditions for the game termination are derived. An explicit form of the guaranteed time is found in one specific case.

This chapter suggests that a general scheme for investigation of conflict-controlled processes, illustrates on various types of functional-differential systems. Employment of the Extremal Targetting Rule, developed in, allows for the game termination in the "first absorption" time in the regular and regularized cases. Similar result was obtained by B. N. Pshenichnyi with the use of the convex analysis technique. On the one hand, this method is the result of extension of the Pontryagin Maximum Principle to the game problems. Positional conflict control by the systems of integral and integro-differential equations was studied in the papers of V. L. Pasikov, G.Ts. Chikrii and K. Volyanskij. Investigations of M. S. Gabrielyan and A. V. Kryazhimskii are devoted to the study of positional conflict counteraction of controlled objects groups. It should be noted that, the Extremal Targetting Rule, as applied to linear systems, is based on using the apparatus of support functions, the notion of Aumann integral of set-valued mapping and the Lyapunov theorem on vector measures.

In the development of ideas of B. N. Pshenichnyi, the paper considers a linear differential game of approach with impulse controls. A research technique is proposed, which is based on time extension and oriented to the case where the classical Pontryagin condition does not hold. Sufficient conditions for the finiteness of the guaranteed approach time are obtained. An illustrative example is given.

It was ascertained that evolutionary game of approach with variable delay of information is equivalent to certain game with complete information, but with another dynamics and terminal set. This fact serves as substantiation of the stated in the paper principle of time stretching, which extends considerably the sphere of usage of the first Pontryagin direct method. For illustration of the suggested technique we consider model example of differential game of soft approach of two objects of the second order with different dynamics.

The paper concerns conflict-controlled processes of general kind with a cylindrical terminal set. Solutions of a dynamic system are presented in a general form, encompassing, in particular, processes with various-type fractional derivatives, impulse processes, and systems of integral, integro-differential and difference–differential equations. Ideas of the method of resolving functions are used as a basis for investigation. While scalar resolving functions execute attraction of sets to the origin, the matrix functions introduced in the paper also admit rotation through any angle, which essentially extends the scope of applications of the method. Sufficient conditions for the termination of the game in a guaranteed time in the class of quasistrategies and stroboscopic strategies are developed.

The concept of matrix resolving function is introduced to study dynamic game problems. The sufficient conditions are derived ensuring the possibility for the pursuer to bring the trajectory of a conflict-controlled process to the terminal set. The cases of using quasi-strategies and counter-controls by the pursuer are analyzed separately. Guaranteed times of the game termination for different method's schemes are compared. The theoretical results are illustrated with a model example of "simple motions" on a plane.

We solve the problem of approach of two controlled systems, which perform damped oscillations. To this end we use modification of the first Pontryagin direct method, which is based on construction of pursuer control by the past control of evader. Sufficient conditions about parameters of the game, which guarantee potential of this approach in finite time and for arbitrary initial states, are obtained.

An approach to solving linear differential pursuit games is substantiated. It consists in generating the pursuer’s control based on the evader’s previous behavior. The results are illustrated with model soft-meeting problems.

A general method for solving game problems of pursuit is proposed for dynamical systems with Volterra evolution. The method makes use of resolving functions [1] and the tools of the theory of set-valued mappings. The scheme proposed covers a wide range of functional-differential systems, such as integral, integrodifferential and differential-difference systems of equations defining the dynamics of conflict-controlled processes. A more detailed study is made of game problems for systems with Riemann-Liouville fractional derivatives and regularized Dzhrbashyan-Nersesyan derivatives (“fractal” games). Asymptotic representations of generalized Mittag-Löffler functions are used in the context of the method to establish sufficient conditions for the solvability of game problems.

A game problem of pursuit for a dynamic process described by a system of integro-differential equations is considered. Sufficient conditions are derived for its termination in a class of positional controls. Model examples illustrate the results obtained.

In the paper game problem of approach for the system given in linear Volterra equation is considered. Certain sufficient conditions are obtained to provide existence of solutions of the problem in a class of positional strategies. A case of separate movements is investigated. Relations between differential and integral games are considered. Results are illustrated on a model example.

The presented results concern linear differential games of pursuit with information time lag on the availability of a current state vector. They encompass three kinds of time lag: constant, function of time, and special function of position, vanishing as an object's trajectory approaches the terminal set. As basis for investigation of such games serves the approach, advanced by the author, which consists in reduction of the original game with information time delay to equivalent one with perfect information. The sufficient conditions for the termination of linear differential pursuit game with variable information time lag are derived.

The problem of searching for a fixed target by a controlled object whose motion is governed by a system of ordinary differential equations or by a linear discrete system with a given probability density distribution of the initial position. The necessary conditions for the optimality of the control which maximizes the probability that the object's trajectory will reach the given target set after a fixed time, are determined.