The problem of simple pursuit of two evaders by a group of pursuers on a given time scale is considered in a finite-dimensional Euclidean space. It is assumed that both evaders use the same control and do not leave a convex polyhedral set. The pursuers use counter-strategies based on information about the initial positions and control history of the evaders. For each of the participants, the set of admissible controls is the unit ball centered at zero and the target set is the origin. The goal of the pursuer group is to capture at least one evader by two pursuers or to capture two evaders. A sufficient capture condition is obtained in terms of initial positions and game parameters. The study is based on the method of resolving functions, which yields sufficient conditions for the solvability of the approach problem in some guaranteed time.

In finite-dimensional Euclidean space, a problem of pursuit of two evaders by a group of pursuers, which is described by a linear nonstationary system of differential equations, is considered under the assumption that the fundamental matrix of the homogeneous system is a recurrent function. It is assumed that the evaders use the same control. The pursuers use counterstrategies based on information about the initial positions and the prehistory of the control of the evaders. The set of admissible controls is a strictly convex compact with a smooth boundary, and the goal sets are the origin of coordinates. The goal of the group of pursuers is to catch at least one evader by two pursuers. In terms of initial positions and parameters of the game, a sufficient condition for capture is obtained. This study is based on the method of resolving functions, which makes it possible to obtain sufficient conditions for solvability of the problem of pursuit in some guaranteed time.

The linear problem of pursuing one evader by a group of pursuers is considered in a finite-dimensional Euclidean space. In a given timescale, the problem is described by a linear system with a simple matrix. The set of admissible controls for each participant is the unit ball centered at the origin. The terminal sets are given convex compact sets. The pursuers use counter-strategies based on information about the initial positions and control history of the evader. Sufficient conditions for the capture of the evader by a given number of pursuers are obtained in terms of the initial positions and parameters of the game. Sufficient evasion conditions are obtained for discrete time scales.

In finite-dimensional Euclidean space, we study the problem of a simple pursuit of two evaders by a group of pursuers in a given time scale.It is assumed that the evaders use the same control and do not move out of a convex polyhedral set. The pursuers use counterstrategies based on information on the initial positions and on the prehistory of the control of evaders. The set of admissible controls of each of the participants is a sphere of unit radius with its center at the origin, and the goal sets are the origin. The goal of the group of pursuers is the capture of at least one evader by two pursuers. In terms of the initial positions and parameters of the game, a sufficient condition for capture is obtained. The study is based on the method of resolving functions, which makes it possible to obtain sufficient conditions for solvability of the pursuit problem in some guaranteed time.

We consider the generalized nonstationary Pontryagin example for a game with many players and the same dynamic and inertial capabilities of the players. We obtain sufficient conditions for the multiple capture of a given number of evaders by a group of pursuers, provided that the evaders use program strategies and each of the pursuers captures at most one evader.

In finite-dimensional Euclidean space, we address the problem of simple pursuit of a group of evaders by a group of pursuers in a given time scale with equal opportunities for all participants. The set of controls of each participant is a sphere of unit radius with its center at the origin. The goal of the group of pursuers is to catch all evaders. The goal sets are the origin. The goal of the evaders is the opposite one, namely, for at least one evader to avoid capture. Conditions for solvability of the local and global problems of evasion and the upper and lower estimates of the minimal number of evaders avoiding a given number of pursuers from any initial positions are obtained.

In finite-dimensional Euclidean space, an analysis is made of the problem of pursuit of two evaders by a group of pursuers, which is described by a linear nonstationary system of differential equations, under the assumption that the fundamental matrix of the homogeneous system is a recurrent function. It is assumed that the evaders use the same control. The pursuers use counterstrategies based on information about the initial positions and the prehistory of the control of the evaders. The set of admissible controls is a strictly convex compact with a smooth boundary, and the goal sets are the origin of coordinates. The goal of the group of pursuers is the capture of at least one evader by two pursuers or the capture of two evaders. In terms of the initial positions and parameters of the game, a sufficient condition for capture is obtained. This study is based on the method of resolving functions, which makes it possible to obtain sufficient conditions for solvability of the problem of pursuit in some guaranteed time.

In a finite-dimensional Euclidean space, we consider the problem of pursuit by a group of pursuers of one evader, which is described by a system of equations with a Caputo derivative of order α , where the sets of feasible controls are convex compact sets. We obtain sufficient conditions for the solvability of pursuit and evasion problems, in the study of which the method of resolving functions is used.

In a finite-dimensional Euclidean space, we consider the problem of pursuit by a group of pursuers of one evader, which is described by a system of equations with a Caputo derivative of order a , where the sets of feasible controls are convex compact sets. We obtain sufficient conditions for the solvability of pursuit and evasion problems, in the study of which the method of resolving functions is used.

A problem of pursuit of one evader by a group of pursuers is considered in a finite-dimensional Euclidean space. The dynamics is described by the system $$D^{(\alpha_{i})}z_{i}=A_{i}z_{i}+B_{i}u_{i}-C_{i}v,$$ $$u_{i}\in U_{i},$$ $$v\in V,$$ where $$D^{(\alpha)}f$$ is the Caputo derivative of order  $$\alpha$$ of a function  $$f$$ . The sets of admissible controls of the players are convex and compact. The terminal set consists of cylindrical sets  $$M_{i}$$ of the form $$M_{i}=M_{i}^{1}+M_{i}^{2}$$ , where $$M_{i}^{1}$$ is a linear subspace of the phase space and $$M_{i}^{2}$$ is a convex compact set from the orthogonal complement of  $$M_{i}^{1}$$ . We propose two approaches to solving the problem, which ensure the termination of the game in a certain guaranteed time in the class of quasi-strategies. In the first approach, the pursuers construct their controls so that the terminal sets “cover” the evader’s uncertainty region. In the second approach, the pursuers construct their controls using resolving functions. The theoretical results are illustrated by model examples.

In a finite-dimensional Euclidean space, the problem of pursuit by a group of pursuers of two evaders described by a system of the form $$\dot z_{ij} = u_i - v,\quad u_i, v \in V $$ is considered. It is assumed that all evaders use the same control. The pursuers use counterstrategies based on information about the initial positions and control history of the evaders. The set of admissible controls $V$ is unit ball centered at zero, target sets are the origin. The goal of the pursuers' group is to capture at least one evader by two pursuers or to capture two evaders. In terms of initial positions and game parameters a sufficient condition for the capture is obtained. In the study, the method of resolving functions is used as a basic one, which allows obtaining sufficient conditions for the solvability of the approach problem in some guaranteed time.

The problem of conflict interaction between a group of pursuers and a group of evaders in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players is described by a system of differential equations with fractional derivatives and a simple matrix. The target sets are the origin. The aim of the group of pursuers is to capture at least one evader. Counterstrategies are acceptable strategies for pursuers. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evaders. The method of resolving functions is used to solve the problem, which is used in differential games of pursuit by a group of pursuers of one evader.

In a finite-dimensional Euclidean space, the problem of pursuing one evader by a group of pursuers is considered, described by a system of the form D-(alpha) z(i) = alpha(i)z(i) + u(i) - upsilon, u(i,) upsilon is an element of V, where D-(alpha) f is the Caputo derivative of order alpha is an element of (0, 1) of the function f. The set of admissible controls V is a convex compact, alpha(i) are non-positive real numbers. The aim of the group of pursuers is to capture the evader. The terminal sets are the origin of coordinates. Sufficient conditions for catching one evader in the class of quasi-strategies are obtained. Using quasi-strategies in an auxiliary game, sufficient conditions for catching an evader in the class of positional strategies with a guide are obtained.

In finite-dimensional Euclidean space, we analyze the problem of pursuit of a single evader by a group of pursuers, which is described by a system of differential equations with Caputo fractional derivatives of order \(alpha.\) The goal of the group of pursuers is the capture of the evader by at least \(m\) different pursuers (the instants of capture may or may not coincide). As a mathematical basis, we use matrix resolving functions that are generalizations of scalar resolving functions. We obtain sufficient conditions for multiple capture of a single evader in the class of quasi-strategies. We give examples illustrating the results obtained.

УДК 517.977 MSC: 49N79, 49N70, 91A24 DOI: 10.21538/0134-4889-2022-28-3-129-141 Работа выполнена при поддержке РНФ (проект 21-71-10070). В конечномерном евклидовом пространстве рассматривается задача преследования группой преследователей одного убегающего, описываемая системой \begin{gather*} D^{(\alpha_i)}z_i = A_iz_i + B_iu_i - C_iv, \quad u_i\in U_i,\quad v\in V,

The paper is based on the results of the fieldwork carried out in 2020 by staff of the Department for Multidisciplinary Scientific Research and six laboratories of severalKarelian Research Centre’s institutes (of Biology; Geology; Forest Research; Linguistics, Literature and History) in the central part of the Republic of Karelia, in Muezersky District. In addition, an extensive, partially published research background from previous studies was used. The work resulted in a synthesis of the data that characterize and substantiate the designation of a cluster-type protected area (PA) of regional significance. It is made up of the following three clusters: 1) landscape nature reserve “Low-mountain landscapes of the West-Karelian Upland” with 11 900 ha; 2) landscape nature monument “Lake Pizanets” with 407.3 ha; 3) landscape nature monument “Mount Vottovaara” with 1 600 ha. The information about these sites is arranged in the following order: 1) geographic location; 2) overall conservation and recreational value as a ground for the designation; 3) key characteristics of the ecosystems (geological-geomorphological, hydrographic, edaphic, of landscapes, mires, forests, flora and fauna, recreational qualities); 4) proposals on the boundaries and size of the protected area. Essentially, the article is a summary of all the data available so far that characterize the ecosystems of central Karelia that have the highest conservation and recreational value. The triple-cluster system of the planned PA, with its clusters situated some 20–50 km apart within a single triangular outline encompassing waterside protection areas, is regarded as an entity.

A problem of pursuit of one or several evaders by a group of pursuers is considered in a finite-dimensional Euclidean space. The problem is described by the system(z) over dot = A(ij)z(ij) + u(i) - v(i), u(i) is an element of U-i, v(j) is an element of V-j.The aim of the group of pursuers is to capture at least q evaders, where each evader must be captured by at least m different pursuers; the capture moments may be different. The terminal sets are the origin. Matrix resolving functions, which generalize scalar resolving functions, are used as a mathematical basis. Sufficient conditions for the multiple capture of one evader in the class of quasi-strategies are obtained. Under the assumption that the evaders use program strategies and each pursuer captures at most one evader, sufficient conditions for the solvability of the problem on the multiple capture of a given number of evaders are obtained in terms of the initial positions. Hall's theorem on a system of distinct representatives is used to prove the main theorem. Examples are given to illustrate the obtained results.

The problem of pursuing a group of evaders by a group of pursuers with equal opportunities for all participants is considered in the finite-dimensional Euclidean space. It is assumed that the movement of the players is simple in a given timescale. Additionally, we assume that in the process of the game each evader does not move out of a convex set with a nonempty interior. The goal of the pursuers' group is to capture at least q evaders, and each evader must be captured by at least r different pursuers, and the moments of capture may not coincide. Terminal sets are the coordinates origin. Assuming that the evaders use program strategies, and each pursuer catches no more than one evader in terms of initial positions, sufficient conditions for the solvability of the pursuit problem are obtained. In the research, the method of resolving functions is used as the base method, which makes it possible to obtain sufficient conditions for the solvability of the approach problem with one evader in some guaranteed time. To prove the main theorem, Hall's theorem of the system of various representatives is used.

In finite-dimensional Euclidean space, an analysis is made of the problem of pursuit of a single evader by a group of pursuers, which is described by a system of the form [Formula: see text] The goal of the group of pursuers is the capture of the evader by no less than [Formula: see text] different pursuers (the instants of capture may or may not coincide). Matrix resolving functions, which are a generalization of scalar resolving functions, are used as a mathematical basis of this study. Sufficient conditions are obtained for multiple capture of a single evader in the class of quasi-strategies. Examples illustrating the results obtained are given.