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Bahrom Samatov

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Родилься в 1957 году в Узбекистане. 1991 году получил степень кандидата физико- математических наук, 2016 доктора физико- математических наук. Настоящие время ведущий научный сотрудник институту математики им. В.И.Романовского Узбекистана.

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Papers by Bahrom Samatov

The Pursuit–Evasion problems in a differential game with GGr-constraints on controls

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

In the present paper, we consider pursuit and evasion problems in a simple motion differential ga... more In the present paper, we consider pursuit and evasion problems in a simple motion differential game when Pursuer's control is subjected to geometric constraint and Evader's control is subjected to Grönwall type constraint. In order to solve the pursuit problem, the parallel convergence strategy (the Πstrategy) for the Pursuer is constructed, and sufficient conditions of pursuit are obtained. Also, we prove that the Π-strategy is an optimal strategy of Pursuer. In solving of the evasion problem, we propose an admissible control function to the Evader, and we obtaine sufficient conditions of evasion. In addition, an estimation of the distance between the objects during the game is provided.

On the Life-Line Game of the Inertial Players with Integral and Geometric Constraints

In this paper, we consider a pursuit-evasion game of inertial players, where pursuer’s control is... more In this paper, we consider a pursuit-evasion game of inertial players, where pursuer’s control is subject to integral constraint, and evader’s control is subject to geometric constraint. In the pursuit problem, the main tool is the strategy of parallel pursuit. Sufficient conditions are obtained for the solvability of pursuit-evasion problems. Also, the main lemma describing the monotonicity of an attainability domain of the evader is proved and an explicit analytical formula for this domain is given. As one of the important results of our paper, the Isaacs Life-line game is solved for a special case.

Pursuit-evasion differential games with Gr-constraints on controls

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta

In the paper, a pursuit-evasion differential game is considered when controls of the players are ... more In the paper, a pursuit-evasion differential game is considered when controls of the players are subject to differential constraints in the form of Grönwall's integral inequality. The strategy of parallel pursuit (briefly, $\Pi$-strategy) was introduced and used by L.A. Petrosyan to solve simple pursuit problems under phase constraints on the states of the players in the case when control functions of both players are chosen from the class $L_\infty$. In the present work, the $\Pi$-strategy is constructed for a simple pursuit problem in the cases when control functions of both players are chosen from different classes of the Grönwall type constraints, and sufficient conditions of capture and optimal capture time are found in these cases. To solve the evasion problem, we suggest a control function for the Evader and find sufficient conditions of evasion. In addition, an attainability domain of the players is constructed and its conditions of embedding in respect to time are given...

Differential Games with Inertial Players Under the Langenhop Type Constraints on Controls

Lobachevskii journal of mathematics, Oct 1, 2023

Problems of Group Pursuit with Integral Constraints on Controls of the Players. II1

Cybernetics and Systems Analysis, 2013

ABSTRACT

About One of the Problem Isaacs for LG Game

Scientific Bulletin of Namangan State University, 2020

In this article is considered the problem of R. Isaac's "Life-line", when liner constraint is imp... more In this article is considered the problem of R. Isaac's "Life-line", when liner constraint is imposed on the pursuer control class which is a generalization of integral as well as geometric constraints and only a geometric constraint is imposed on the evader control class and here it is studied for win of the pursuer.

The Strategy of Parallel Pursuit Fordifferential Game of the First Order Withgronwall-Bellman Constraints

Scientific Bulletin of Namangan State University, 2020

A method of solvable functions for solution of pursue problem at integral restriction on controls

Cornell University - arXiv, Jul 30, 2012

The Strategy of Parallel Pursuit for Differential Game of the Second Order

In this work is considered a differtial game of the second order, when control functions of the p... more In this work is considered a differtial game of the second order, when control functions of the players satisfies geometric constraints. The proposed method substantiates the parallel approach strategy in this differential game of the second order. The new sufficient solvability conditions are obtained for problem of the pursuit.

The Resolving Functions Method for the Pursuit Problem with Integral Constraints on Controls

Journal of Automation and Information Sciences, 2013

On a pursuit-evasion problem under a linear change of the pursuer resource

Siberian Advances in Mathematics, 2013

We study a pursuit-evasion problem in the case when an integral constraint is imposed on the purs... more We study a pursuit-evasion problem in the case when an integral constraint is imposed on the pursuer control class which is a generalization of integral as well as geometric constraints and only a geometric constraint is imposed on the evader control class. We prove the theorem of alternative. The optimal pursuit problem is solved by a generalized parallel pursuit strategy, and lower bounds for the distance between the pursuer and the evader are established in the pursuit problem.

Differential games of the second order

In this paper, we study the pursuit-evasion problem for the second order differential game when t... more In this paper, we study the pursuit-evasion problem for the second order differential game when the initial positions of moving objects are linearly dependent and controls of the players have geometric constraints. The new sufficient solvability conditions are obtained for problems of the pursuit and evasion

Differential Games with the Langenhop Type Constrains on Controls

Lobachevskii Journal of Mathematics

Differential Games of the Second Order with Integralconstraints

Scientific Bulletin of Namangan State University, 2020

In this paper, we study the pursuit problem for differential game of the second order with integr... more

On Evasion Problem for the Case Which Imposedgeometricalgronwall Constraint

Scientific Bulletin of Namangan State University, 2020

Differential Game with a Lifeline for the Inertial Movements of Players

Ural Mathematical Journal, 2021

In this paper, we study the well-known problem of Isaacs called the "Life line" game wh... more In this paper, we study the well-known problem of Isaacs called the "Life line" game when movements of players occur by acceleration vectors, that is, by inertia in Euclidean space. To solve this problem, we investigate the dynamics of the attainability domain of an evader through finding solvability conditions of the pursuit-evasion problems in favor of a pursuer or an evader. Here a pursuit problem is solved by a parallel pursuit strategy. To solve an evasion problem, we propose a strategy for the evader and show that the evasion is possible from given initial positions of players. Note that this work develops and continues studies of Isaacs, Petrosjan, Pshenichnii, Azamov, and others performed for the case of players' movements without inertia.

A Linear Differential Game with Gronwall Type Constraint

In the paper, it has been investigated how Gronwall`s inequality can be used in theory of Differe... more In the paper, it has been investigated how Gronwall`s inequality can be used in theory of Differential Games. Here a pursuit problem of linear differential game has been studied when constraints of Gronwall type generalizing geometrical constraint imposed on control parameters of players. To solve the pursuit problem it will be proposed a parallel pursuit strategy (-strategy) and its structure will be studied in relation to parameters. In this article, the authors` previous works will be advanced and extended. As a result of the work, the new sufficiency conditions will be suggested.

Evasion Problem in Linear Differential Game with Gronwall Type Constraint

The main aim of this work is to present some natural applications of Gronwall type inequalities i... more The main aim of this work is to present some natural applications of Gronwall type inequalities in the Differential Games. In the present, the evasion problem is studied in linear differential games when Gronwall type constraints imposed on control functions of players. The Gronwall type constraint generalizes geometrical constraint. To solve the evasion problem, we propose a particular strategy for evader and study its structure depending on the parameters. This work develops and extends the ideas of works of Isaacs, Petrosyan, Pshenichnii and other researchers, including the author. Here the new sufficient solvability conditions for evader will be proposed.

The Problem of Ramchundra for a Problem of -Capture

We study the problem for a differential game of lcapture. We study the problem of Ramchundra with... more

Pursuit-Evasion Differential Games with the Grönwall Type Constraints on Controls

Ural Mathematical Journal, 2020

A simple pursuit-evasion differential game of one pursuer and one evader is studied. The players&... more A simple pursuit-evasion differential game of one pursuer and one evader is studied. The players' controls are subject to differential constraints in the form of the integral Grönwall inequality. The pursuit is considered completed if the state of the pursuer coincides with the state of the evader. The main goal of this work is to construct optimal strategies for the players and find the optimal pursuit time. A parallel approach strategy for Grönwall-type constraints is constructed and it is proved that it is the optimal strategy of the pursuer. In addition, the optimal strategy of the evader is constructed and the optimal pursuit time is obtained. The concept of a parallel pursuit strategy (\(\Pi\)-strategy for short) was introduced and used to solve the quality problem for "life-line" games by L.A.Petrosjan. This work develops and expands the works of Isaacs, Petrosjan, Pshenichnyi, and other researchers, including the authors.

The Pursuit–Evasion problems in a differential game with GGr-constraints on controls

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

In the present paper, we consider pursuit and evasion problems in a simple motion differential ga... more In the present paper, we consider pursuit and evasion problems in a simple motion differential game when Pursuer's control is subjected to geometric constraint and Evader's control is subjected to Grönwall type constraint. In order to solve the pursuit problem, the parallel convergence strategy (the Πstrategy) for the Pursuer is constructed, and sufficient conditions of pursuit are obtained. Also, we prove that the Π-strategy is an optimal strategy of Pursuer. In solving of the evasion problem, we propose an admissible control function to the Evader, and we obtaine sufficient conditions of evasion. In addition, an estimation of the distance between the objects during the game is provided.

On the Life-Line Game of the Inertial Players with Integral and Geometric Constraints

In this paper, we consider a pursuit-evasion game of inertial players, where pursuer’s control is... more In this paper, we consider a pursuit-evasion game of inertial players, where pursuer’s control is subject to integral constraint, and evader’s control is subject to geometric constraint. In the pursuit problem, the main tool is the strategy of parallel pursuit. Sufficient conditions are obtained for the solvability of pursuit-evasion problems. Also, the main lemma describing the monotonicity of an attainability domain of the evader is proved and an explicit analytical formula for this domain is given. As one of the important results of our paper, the Isaacs Life-line game is solved for a special case.

Pursuit-evasion differential games with Gr-constraints on controls

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta

In the paper, a pursuit-evasion differential game is considered when controls of the players are ... more In the paper, a pursuit-evasion differential game is considered when controls of the players are subject to differential constraints in the form of Grönwall's integral inequality. The strategy of parallel pursuit (briefly, $\Pi$-strategy) was introduced and used by L.A. Petrosyan to solve simple pursuit problems under phase constraints on the states of the players in the case when control functions of both players are chosen from the class $L_\infty$. In the present work, the $\Pi$-strategy is constructed for a simple pursuit problem in the cases when control functions of both players are chosen from different classes of the Grönwall type constraints, and sufficient conditions of capture and optimal capture time are found in these cases. To solve the evasion problem, we suggest a control function for the Evader and find sufficient conditions of evasion. In addition, an attainability domain of the players is constructed and its conditions of embedding in respect to time are given...

Differential Games with Inertial Players Under the Langenhop Type Constraints on Controls

Lobachevskii journal of mathematics, Oct 1, 2023

Problems of Group Pursuit with Integral Constraints on Controls of the Players. II1

Cybernetics and Systems Analysis, 2013

ABSTRACT

About One of the Problem Isaacs for LG Game

Scientific Bulletin of Namangan State University, 2020

In this article is considered the problem of R. Isaac's "Life-line", when liner constraint is imp... more In this article is considered the problem of R. Isaac's "Life-line", when liner constraint is imposed on the pursuer control class which is a generalization of integral as well as geometric constraints and only a geometric constraint is imposed on the evader control class and here it is studied for win of the pursuer.

The Strategy of Parallel Pursuit Fordifferential Game of the First Order Withgronwall-Bellman Constraints

Scientific Bulletin of Namangan State University, 2020

A method of solvable functions for solution of pursue problem at integral restriction on controls

Cornell University - arXiv, Jul 30, 2012

The Strategy of Parallel Pursuit for Differential Game of the Second Order

In this work is considered a differtial game of the second order, when control functions of the p... more In this work is considered a differtial game of the second order, when control functions of the players satisfies geometric constraints. The proposed method substantiates the parallel approach strategy in this differential game of the second order. The new sufficient solvability conditions are obtained for problem of the pursuit.

The Resolving Functions Method for the Pursuit Problem with Integral Constraints on Controls

Journal of Automation and Information Sciences, 2013

On a pursuit-evasion problem under a linear change of the pursuer resource

Siberian Advances in Mathematics, 2013

We study a pursuit-evasion problem in the case when an integral constraint is imposed on the purs... more We study a pursuit-evasion problem in the case when an integral constraint is imposed on the pursuer control class which is a generalization of integral as well as geometric constraints and only a geometric constraint is imposed on the evader control class. We prove the theorem of alternative. The optimal pursuit problem is solved by a generalized parallel pursuit strategy, and lower bounds for the distance between the pursuer and the evader are established in the pursuit problem.

Differential games of the second order

In this paper, we study the pursuit-evasion problem for the second order differential game when t... more In this paper, we study the pursuit-evasion problem for the second order differential game when the initial positions of moving objects are linearly dependent and controls of the players have geometric constraints. The new sufficient solvability conditions are obtained for problems of the pursuit and evasion

Differential Games with the Langenhop Type Constrains on Controls

Lobachevskii Journal of Mathematics

Differential Games of the Second Order with Integralconstraints

Scientific Bulletin of Namangan State University, 2020

In this paper, we study the pursuit problem for differential game of the second order with integr... more

On Evasion Problem for the Case Which Imposedgeometricalgronwall Constraint

Scientific Bulletin of Namangan State University, 2020

Differential Game with a Lifeline for the Inertial Movements of Players

Ural Mathematical Journal, 2021

In this paper, we study the well-known problem of Isaacs called the "Life line" game wh... more In this paper, we study the well-known problem of Isaacs called the "Life line" game when movements of players occur by acceleration vectors, that is, by inertia in Euclidean space. To solve this problem, we investigate the dynamics of the attainability domain of an evader through finding solvability conditions of the pursuit-evasion problems in favor of a pursuer or an evader. Here a pursuit problem is solved by a parallel pursuit strategy. To solve an evasion problem, we propose a strategy for the evader and show that the evasion is possible from given initial positions of players. Note that this work develops and continues studies of Isaacs, Petrosjan, Pshenichnii, Azamov, and others performed for the case of players' movements without inertia.

A Linear Differential Game with Gronwall Type Constraint

In the paper, it has been investigated how Gronwall`s inequality can be used in theory of Differe... more In the paper, it has been investigated how Gronwall`s inequality can be used in theory of Differential Games. Here a pursuit problem of linear differential game has been studied when constraints of Gronwall type generalizing geometrical constraint imposed on control parameters of players. To solve the pursuit problem it will be proposed a parallel pursuit strategy (-strategy) and its structure will be studied in relation to parameters. In this article, the authors` previous works will be advanced and extended. As a result of the work, the new sufficiency conditions will be suggested.

Evasion Problem in Linear Differential Game with Gronwall Type Constraint

The main aim of this work is to present some natural applications of Gronwall type inequalities i... more The main aim of this work is to present some natural applications of Gronwall type inequalities in the Differential Games. In the present, the evasion problem is studied in linear differential games when Gronwall type constraints imposed on control functions of players. The Gronwall type constraint generalizes geometrical constraint. To solve the evasion problem, we propose a particular strategy for evader and study its structure depending on the parameters. This work develops and extends the ideas of works of Isaacs, Petrosyan, Pshenichnii and other researchers, including the author. Here the new sufficient solvability conditions for evader will be proposed.

The Problem of Ramchundra for a Problem of -Capture

We study the problem for a differential game of lcapture. We study the problem of Ramchundra with... more

Pursuit-Evasion Differential Games with the Grönwall Type Constraints on Controls

Ural Mathematical Journal, 2020

A simple pursuit-evasion differential game of one pursuer and one evader is studied. The players&... more A simple pursuit-evasion differential game of one pursuer and one evader is studied. The players' controls are subject to differential constraints in the form of the integral Grönwall inequality. The pursuit is considered completed if the state of the pursuer coincides with the state of the evader. The main goal of this work is to construct optimal strategies for the players and find the optimal pursuit time. A parallel approach strategy for Grönwall-type constraints is constructed and it is proved that it is the optimal strategy of the pursuer. In addition, the optimal strategy of the evader is constructed and the optimal pursuit time is obtained. The concept of a parallel pursuit strategy (\(\Pi\)-strategy for short) was introduced and used to solve the quality problem for "life-line" games by L.A.Petrosjan. This work develops and expands the works of Isaacs, Petrosjan, Pshenichnyi, and other researchers, including the authors.