Simple Pursuit Problem Research Articles

Simple Pursuit Problem with Phase Constraints of Two Coordinated Evaders on Time Scales

Simple Pursuit Problem with Phase Constraints of Two Coordinated Evaders on Time Scales

The differential game “Cossacks–robbers” on time scales

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.

Two-time capture of coordinated evaders in a simple pursuit problem

In a finite-dimensional Euclidean space, the problem of pursuit of two evaders by a group of pursuers 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 a unit ball centered at zero, target sets are the origin of coordinates. The goal of the pursuers' group is to capture at least one evader by two pursuers. 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.

Simple pursuit problem with phase restrictions of two coordinated evaders on time scales

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

Pursuit-evasion differential games with Gr-constraints on controls

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. Results of this work continue and extend the works of R. Isaacs, L.A. Petrosyan, B.N. Pshenichnyi, A.A. Chirii, A.A. Azamov and other researchers, including the authors.

On one simple pursuit problem of two rigidly coordinated evaders

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.

Multiple Capture of a Given Number of Evaders in the Problem of Simple Pursuit with Phase Restrictions on Timescales

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.

Geometric modeling of the parallel approach method in some transport problems

This article presents a quasidiscrete geometric model based on the problem of simple pursuit on a plane by the method of parallel approximation. Apollonius and the characteristic lines associated with it in this geometric model for a predetermined trajectory are in the optimal path of the pursuer. The geometric model is also used in solving some transport problems: visualization, construction of calculation grids, etc. Modeling was performed in the computer mathematics system MathCAD. Based on the simulation results, an animation clip was created, where you can view the movement and transformation of the Apollonius circle and associated characteristic points and lines.

Оценка времени преследования в дифференциальных играх многих игроков на выпуклом компакте

Статья посвящена изучению задачи построения стратегии преследования в простых дифференциальных играх многих лиц с фазовыми ограничениями в состоянии игроков, в смысле попадания в некоторый окрест беглеца. Игра происходит в n-мерном евклидовом пространстве на выпуклом компакте. Рассматривается задача преследования, когда число преследователей n–1, то есть меньше n, в смысле — l-поимка. Предложена структура построения управлений преследования, которая обеспечит завершение игры за конечное время. Получена оценка сверху времени игры для завершения преследования. Рассмотрена вспомогательная задача простого преследования на единичном кубе в первом ортанте и построена стратегия преследователей для завершения игры с особыми начальными позициями. Полученные результаты применяются для решения дифференциальных игр с произвольными начальными позициями. Для этой задачи предложена структура построения стратегии преследования, которая обеспечит завершение игры за конечное время. Также рассматривается обобщение задачи в смысле усложнения помехи. Рассматривается более общая задача простого преследования на кубе произвольного размера в первом ортанте. С помощью предложенных стратегий доказаны возможности завершения преследования и получена оценка времени. Как следствие этого результата, получены оценки снизу и сверху для времени преследования в игре с препятствиями типа шара. Получены оценки времени преследования, когда компакт – произвольно выпуклое множество. Определено понятие выпуклого множества относительно направления сечения, которое необязательно выпукло. В нем изучена задача простого преследования в дифференциальной игре многих игроков и показаны возможности завершения преследования с применением предложенной стратегии. Оценивается сверху время завершения преследования данной игры.

ВИЗУАЛИЗАЦИЯ ОКРУЖНОСТЕЙ АПОЛЛОНИЯ ПРИ ГЕОМЕТРИЧЕСКОМ МОДЕЛИРОВАНИИ МЕТОДА ПАРАЛЛЕЛЬНОГО СБЛИЖЕНИЯ НА ПЛОСКОСТИ

This article presents a quasidiscrete geometric model of the problem of simple pursuit on a plane by the method of parallel approximation. For each moment of time, the Apollonius circle and associated characteristic lines are constructed. In this geometric model, for the predetermined target path, the optimal path of the pursuer is found. Modeling was performed in the computer mathematics system MathCAD. Based on the simulation results, an animated video was made, where you can view the movement and transformation of the Apollonius circle and associated characteristic points and lines.

On the “equivalence” of two evasion problems with multiple evaders

A simple pursuit problem in which a group of pursuers and a group of evaders are involved is considered under the condition that the group of pursuers includes participants whose maximum speed coincides with that of the evaders and participants whose maximum speed is strictly lower than the maximum speed of the evaders. The aim of the group of pursuers is to capture all the evaders. The aim of the group of evaders is prevent the capture, that is, to allow at least one of the evaders to avoid the rendezvous. It is shown that, if in the game in which all the participants have equal capabilities at least one of the evaders avoids the rendezvous on an infinite time interval, then as a result of the addition of any number of pursuers with less capabilities, at least one of the evaders will avoid the rendezvous on any finite time interval.

Differential games with many pursuers when evader moves on the surface of a cylinder

We study a pursuit differential game with many Pursuers when the Evader moves on the surface of a given cylinder. Maximal speeds of all players are equal. We consider two cases: in the first case, the Pursuers move arbitrarily without phase constraints; and in the second case, the Pursuers move on the surface of the cylinder. In both cases, we give necessary and sufficient conditions to complete the pursuit. In addition, in the second case, we show that pursuit differential game on a cylinder are equivalent to a differential game on the plane with many groups of Pursuers where each group consists of infinite number of pursuers having the same control parameter. References Isaacs, R. Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control, and Optimization. New York: Wiley, 1963. Petrov, N. N. A problem of group pursuit with phase constraints. J. Appl. Math. Mech. 1988, 52, No 6, 1030--1033. Petrosyan, L. A. Survival differential game with many participants. Dokl. Akad. Nauk USSR. 1965, 161, No 2, 285--287. Petrosyan, L. S. Differentsial'nye igry presledovaniya (Pursuit Differential Games). Leningrad (SPb): Leningr.State Univ(SPbSU), 1977. Pshennichnyi, B. N. Simple pursuit of several targets. Kibernetika. 1976, No 3, 145--146. Chernous'ko, F. L. A problem of evasion of several pursuers. J. Appl. Maths Mechs.1976, 40, No 1, 14--24. Ivanov R. P. Simple pursuit-evasion on a compact. Dokl. Akad. Nauk SSSR. 1980, 254, No 6, 1318--1321. Melikyan, A. A. and Ovakimyan, N. V. Singular trajectories in the problem of simple pursuit on a manifold. J. Appl. Math. Mech. 1991, 55, No 1, 42--48. Melikyan, A. A. and Ovakimyan, N. V. Differential Games of Simple Pursuit and Approach on Manifolds. Institute of Mechanics, National Academy of Sciences of Armenia. Yerevan. Preprint, 1993. Kuchkarov, A. Sh. The problem of optimal approach in locally euclidean spaces. Automation and Remote Control. 2007, 68, No 6, 974-978. Kuchkarov, A. Sh. A simple pursuit--evasion problem on a ball of a Riemannian manifold. Mathematical Notes. 2009, 85, No 2, 190--197. Azamov, A. On a problem of escape along a prescribed curve. J. Appl. Math. Mech. 1982, 46, No 4, 553--555. Kuchkarov, A. Sh. and Rikhsiev, B. B. on the solution of a pursuit problem with phase constraints. Automation and Remote Control. 2001, 62, No 8, 1259--1262. Ibragimov G. I. A game problem on a closed convex set. Siberian advances in mathematics., 2002, 12, No 3, 16--31. Nikulin, V. V. and Shafarevich, I. R. Geometriya i Gruppy (Geometric and Groups). Moscow: Nauka, 1983.

Об одной задаче простого преследования

Об одной задаче простого преследования

Simultaneous multiple capture in a simple pursuit problem

The necessary and sufficient conditions for simultaneous multiple capture in a simple group pursuit problem with different opportunities for the participants are obtained.

On Substantiation of the Proportional Navigation Method in the Simple Pursuit Problem

On Substantiation of the Proportional Navigation Method in the Simple Pursuit Problem

Singular trajectory in the problem of simple pursuit on a manifold

Games with at most two minimal geodesics connecting two points in their phase space (manifold) are considered. It is shown that singular trajectories - envelopes of geodesics - may develop in such games. A necessary condition for the existence of singular motions (non-emptiness of the set (3.5)) is derived as a certain requirement from the geometrical properties of the phase manifold of the game. An algorithm to construct the singular motions is proposed and the Hamiltonian equations describing these motions are given. The sufficiency of the proposed construction is investigated numerically for particular examples. The paper generalizes and extends the previous study /1/.

On applications of singular perturbation techniques in nonlinear optimal control

The technique of singular perturbations (SPT) has been applied with considerable success in several nonlinear optimal control problems. In many cases the zero-order approximation of the optimal control function has been expressed in a feedback form. This paper deals with topics involved in such closed-loop application, which seem to merit further discussion. It is formally demonstrated that a ‘forced’ singular perturbation model (obtained by artificial insertion of the perturbation parameter) results in the same zero-order composite feedback control solution as a classical singularly perturbed model (where a small parameter of physical significance appears as a consequence of a scaling transformation). The accuracy of the zero-order feedback approximation depends in both cases on the actual time scale separation of the variables. Two inherent limitations of the feedback solution are also pointed out: (1) first and higher-order correction terms of the zero-order approximation have to be computed by a predictive or off-line integration; (2) on-line implementation of SPT control strategy in a terminal boundary layer requires iterative computations. A simple pursuit problem serves as an illustrative example.