Differential Games Of Pursuit Research Articles

Some Results on Pursuit Games for an Infinite System of Ternary Differential Equations

This paper aims to study a one-pursuer, one evader pursuit differential game for a higher level of infinite system that is an infinite system of first order ternary differential equations, and prove completion of pursuit in the game. Both integral constraints and geometric constraints are subjected on the players' control functions, thus two separate cases of pursuit games are examined. In the game, the pursuer wants to take the state of the system into the origin of l2 space at some finite time interval, whereas evader avoids this from happening. For every case, we solve the control problem by establishing the admissible control function. In order to achieve the pursuer's objective, we then construct an admissible strategy for the pursuer and develop an equation for the guaranteed pursuit time of the game.

On the problem of controlling a second-order nonlinear system by means of discrete control under disturbance

The problem of bringing a trajectory to a neighborhood of zero under disturbance is considered in terms of a differential pursuit game. The dynamics are described by a nonlinear autonomous system of second-order differential equations. The set of values of the pursuer's controls is finite, and that of the evader (disturbance) is compact. The goal of the control, that is, the goal of the pursuer, is to bring, within a finite time, the trajectory to any predetermined neighborhood of zero, regardless of the actions of the disturbance. To construct the control, the pursuer knows only the phase coordinates and the value of the velocity at some discrete moments of time and the choice of the disturbance control is unknown. Conditions are obtained for the existence of a set of initial positions, from each point of which a capture occurs in the specified sense. Moreover, this set contains a certain neighborhood of zero. The winning control is constructed constructively and has an additional property specified in the theorem. In addition, an estimate of the time required to bring the speed from one given point to the neighborhood of another given point under disturbance conditions was obtained.

Differential Game of Two Pursuers Chasing One Evader with Different Forms of Constraints in l_2- Space

This paper addresses a pursuit differential game involving two Pursuers chasing one Evader within the l2 -space for an infinite system of first-order differential equations. The first Pursuer employs a strategy that satisfied integral constraint, while the second Pursuer uses a strategy governed by a geometric constraint. The goal of each pursuer is to force the state of the system to coincide with a predefined state within a finite time, counteracting the Evader's opposing actions. We construct an explicit strategy to determine the conditions necessary for successful pursuit. Moreover, we explore a control problem involving a single player.

Multi-pursuer pursuit differential game for an infinite system of second order differential equations

We study a pursuit differential game of many pursuers and one evader. The game is described by the infinite systems of $m$ inertial equations. By definition, pursuit in the game is completed if the state and its derivative of one of the systems are equal to zero at some time. In the literature, such a condition of completion of pursuit is also called soft landing. We obtain a condition in terms of energies of players which is sufficient for completion of pursuit in the game. The pursuit strategies are also constructed.

Optimal Pursuit Differential Game Problem for an Infinite System of Binary Differential Equations

Optimal Pursuit Differential Game Problem for an Infinite System of Binary Differential Equations

Solutions to the problem of differential games with electric oscillations by finite difference methods

The article deals with the issues of electromagnetic processes from the point of view of application technology. The problem of voltage normalization in lines in the presence of opposing external forces is studied. The process of realizing the solution of this problem is based on the introduction of a mathematical model of electromagnetic processes occurring in lines as a continuous differential pursuit game described by hyperbolic equations. The finite difference method transforms the continuous pursuit problem into a discrete game. Sufficient conditions are obtained getting into neighbourhood. And with the help of a well-known a priori estimate for the difference between the solutions of a continuous differential and the corresponding discrete game, sufficient conditions are obtained for the possibility of completing the pursuit of a continuous game. A pursuer outputs terminal set, which is voltage normalization.

On Some $$\ell$$-Catch Pursuit Differential Games with Different Players’ Dynamic Equations

On Some $$\ell$$-Catch Pursuit Differential Games with Different Players’ Dynamic Equations

On a Linear Differential Game of Pursuit with Integral Constraints in ℓ2

In this paper, we study the stability, controllability, and differential game of pursuit for an infinite system of linear ODEs in ℓ2. The system we consider has a special right-hand side, which is not diagonal and serves as a toy model for controllable system of infinitely many interacting points. We impose integral constraints on the control parameters. We obtain criteria for stability and null controllability of the system. Further, we construct a strategy for the pursuer that guarantees completion of the pursuit problem for the differential game. To prove controllability we use the so called Gramian operators.

СТОХАСТИЧНА ДЕСКРИПТОРНА ГРА ПЕРЕСЛІДУВАННЯ

A differential pursuit game in a stochastic descriptor linear system is analyzed. The system dynamics is described by Ito’s stochastic differential algebraic equation. Solutions of the equation are presented by the formula of variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct a pursuer control bringing the dynamic vector of the system to a terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transient states in a radio technical filter with random perturbations in the form of white noise. Keywords: stochastic differential algebraic equation, Wiener process, descriptor system, differential game, radio technical filter, white noise.

Sufficient Solvability Conditions for the Problem of Pursuit under an Impulse Action

The article considers a linear differential pursuit game under the condition that an integral constraint is imposed on the evader’s control and the pursuer uses an impulse control. These impulse actions on the object are made at predetermined time points, and the corresponding control is represented using the Dirac delta function. The article studies linear conflicts described by a system of ordinary differential equations whose trajectories have jumps at certain times. The terminal set is represented by a cylinder in an n-dimensional Euclidean space. The problem is solved using the resolving function method. The fact that the lower bound is reached is proved using the theory of support functions. As a result, the quasi-strategy is replaced by an almost stroboscopic strategy and a method for building this strategy is presented. An example of a nonlinear right-hand side is given.

Guaranteed Pursuit and Evasion Times in a Differential Game for an Infinite System in Hilbert Space l2

The present paper is devoted to studying a pursuit differential game described by an infinite system of binary differential equations in Hilbert space l2. The control parameters of the players are subject to geometric constraints. The pursuer tries to bring the state of the system to the origin of the Hilbert space l2, and oppositely, the evader tries to avoid it. Our aim is to construct a strategy for the pursuer to complete a differential game and an evasion control. We obtain an equation for the guaranteed pursuit and evasion times.

$\Pi$-strategy for a differential game of pursuit with integral constraints of a generalized type

The paper investigates a differential game of simple pursuit, when the controls of two opposing players are subject to integral constraints of a generalized type. The generalization of the proposed restriction lies in the fact that it includes previously known restrictions such as integral, geometric, linear, exponential and their mixtures. In general, it includes 25 types of pursuit problems with such different types of constraints. To solve the pursuit problem under such generalized constraints, we propose a parallel pursuit strategy ($\Pi$-strategy for short) and find sufficient conditions for the solvability of this problem. At the end of the article, tables are provided that list each particular type of game, the conditions for its solvability, the resolving function (which determines the corresponding $\Pi$-strategy), and the time of capture.

On some new results in a pursuit differential game with many pursuers and one evader

<abstract><p>We study a simple motion differential game with many pursuers and one evader with equal capabilities in $ \mathbb{R}^n $. The control functions of the players are subject to the Grönwall-type constraints. If the state of the evader coincides with the state of a pursuer, then the game is considered completed. If the state of the evader does not coincide with the state of any pursuer at all times, then we say that evasion is possible. We show that pursuit can be completed if a condition on the convex hull of the initial states of the pursuers is satisfied.</p></abstract>

A differential game of pursuit for an infinite system of simple motion in the plane

In the present paper, we investigate a differential game of pursuit for an infinite system of simple motion in a plane. The control functions of the players satisfies both geometric and integral constraints respectively. In the plane, the game is assume to be completed if the state of the pursuer xk, k = 1, 2, ... is directly coincide with that of the evader yk, k = 1, 2, ..., i.e; xk(x ) = yk(x ), k = 1,2, ..., at some time x and the evader is tries to stop the incident. In addition to that the strategy of the pursuer with respect to geometric and integral constraints will be constructed. Moreover, a numerical example will be given to illustrate the result.

Pursuit Differential Game Problems of L-Catch in A Hilbert Space

We propose some fixed duration pursuit differential game problems of l-catch in a Hilbert space. Players' motions obey ordinary differential equations and the control functions of players are subjected to either integral or geometric constraints. Pursuit is said to be completed in l-catch sense if the distance between the players with conicting goals are less than a fixed constant l. In this sense, we obtain sufficient conditions for the completion of the pursuit and construct a strategy for the pursuer in each of the four different problems considered. Furthermore, we give examples to support our theoretical findings

Positional strategies in fractional order differential games

Problems of the branch of mathematics called differential games, which today find their versatile applications in physics and engineering, are devoted to the construction of chase control when the states of the object are known only at given times in advance. Sufficient conditions for the possibility of completing the pursuit in the sense of hitting a small neighborhood of the terminal set are obtained. A method is indicated for constructing a positional strategy of the pursuer that infers a trajectory to a given neighborhood of the terminal set by the time determined by the first direct method of the theory differential pursuit games. In this case, the chasing party cannot use the fleeing player’s control to build its control; therefore, according to the condition of the problem, the chaser builds its control if it knows the state of the object at the given times in advance.

Differential Game for an Infinite System of Two-Block Differential Equations

We present a pursuit differential game for an infinite system of two-block differential equations in Hilbert space l2. The pursuer and evader control functions are subject to integral constraints. The differential game is said to be completed if the state of the system falls into the origin of l2 at some finite time. The purpose of the pursuer is to bring the state of the controlled system to the origin of the space l2, whereas the evader’s aim is to prevent this. For the optimal pursuit time, we obtain an equation and construct the optimal strategies for the players.

ε-Positional Strategies in Theory of Differential Pursuit Games and Invariance of Constant Multivalued Mappings in Heat-Conductivity Problems

In this paper, we consider two problems. In the first problem, we prove that if the assumption from [1] and one additional condition on the parameters of the game hold, then the pursuit can be completed in any neighborhood of the terminal set. To complete the game, an ε-positional pursuit strategy is constructed.In the second problem, we study the invariance of a given multivalued mapping with respect to the system with distributed parameters. The system is described by the heat-conductivity equation containing additive control terms at the right-hand side.

Pursuit Differential Game with Slow Pursuers on the 1-Skeleton Graph of the Icosahedron

A differential game of m, 3≤m≤6, pursuers and one evader is studied on an icosahedron in R3. All the players move only along the 1-skeleton graph of the icosahedron when the maximal speeds of the pursuers are less than the speed of the evader. Pursuit is said to be completed if the state of a pursuer coincides with the state of evader at some time. We give a sufficient condition of the completion of pursuit in the game.

Complete Solution of a Class of Differential Pursuit Games with Integral Constraint and Impulse Control

Complete Solution of a Class of Differential Pursuit Games with Integral Constraint and Impulse Control