The method of pursuit by several controlled objects of different types

The problem of the pursuit of one evader by several controlled objects of different types is examined. The sufficient conditions are obtained for the pursuit game to terminate in a finite time. The proposed method of pursuer interaction assumes that the pursuing players are separated into two groups, the first of which holds the evader in some domain, while the second searches for the evader in this domain. The paper touches on the researches in /1–9/. Typical examples illustrate the results.     

A game of optimal pursuit of one object by several

A differential game in which m dynamical objects pursue a single one is investigated. All the players perform simple motions. The termination time of the game is fixed. The controls of the first k (k ⩽ m) pursuers are subject to integral constraints and the controls of the other pursuers and the evader are subject to geometric constraints. The payoff of the game is the distance between the evader and the closest pursuer at the instant the game is over. Optimal strategies for the players are constructed and the value of the game is found.     

Differential game of optimal approach of two inertial pursuers to a noninertial evader

In this paper, the game of the optimal approach of two identical inertial pursuers to a noninertial evader is investigated. The duration of the game is fixed. The payoff functional is the distance between the evader and the closest pursuer when the game terminates. The value function is constructed for all possible positions of the game. The regions where the pursuit is one-to-one and the regions where it is essentially collective are described algorithmically. Some analogies between this game and the linear differential game with elliptical vectograms are indicated. It is noted that the focal surface and the dispersal surface are in proximity of one another.     

Successive Pursuit with a Bounded Detection Domain

We study a coplanar model of the successive pursuit of two evaders with unlimited turn rates of the players and a bounded detection domain of a pursuer. Involved in catching the first evader, the pursuer may lose sight of the other. In this case, it must search later for the lost evader in the plane. We describe two guaranteed pursuit strategies obtained as solutions of differential games. Both strategies include a two-stage strategy to shorten to a specified quantity the distance to the nearer evader, and a two-stage strategy to search and capture the other.The strategies are distinguished by their search plans. First, coalition is pursued as a whole. Then, to minimize an uncertainty index, the pursuer approaches the first evader using the strategy of successive pursuit with the unmoved second evader at its last observed position. Subsequently, the pursuer moves directly to that position of the second evader, or according to the more complex plan, alternates between traversing a straight line and arcs of logarithmic spirals. After detection, the remaining evader is captured with the use of a simple pursuit strategy.The barriers fit the strategies. We call them approximate, since they bound the states where the pursuer succeeds with the guaranteed (but not optimal as in the case of ordinary barriers) strategies. These barriers are surfaces of constant values of a special game of degree. The more complex search plan secures a wider winning area.Geometrical interpretations and some numerical results for a set of parameters of the game are provided.     

Isaacs' princess and monster game on the circle

The solution is given of a multistage pursuit-evasion game in which ablind pursuer searches for ablind evader within a finite set of locations arranged in a circle. The players traverse this set by transfering their positions, at each of a succession of instants, from the points which they occupy to ones which are adjacent to them. Some standard results of measure theory are used to construct the players' optimal strategies when the payoff to the evader is the time taken for the pursuer to find him or, more generally, an increasing function of this time.This work was carried out with the support of a CSIRO postgraduate studentship.     

A differential game of approach with two pursuers and one evader

A differential game of approach with one evader and two pursuers with a nonconvex payoff function is considered. The duration of the game is fixed. The payoff functional is the distance between the object being pursued and the pursuer closest to it when the game terminates. An explicit form of the game value is found for all possible game positions. The paper is closely related to Refs. 1–12.The authors would like to thank Professor A. I. Subbotin for his valuable advice and encouragement.     

The game of two identical cars

This paper describes a third-order pursuit—evasion game in which both players have the same speed and minimum turn radius. The game of kind is first solved for thebarrier or envelope of capturable states. When capture is possible, the game of degree is then solved for the optimal controls of the two players as functions of the relative position. The solution is found to include a universal surface for the pursuer and a dispersal surface for the evader.     

An Inexact Spingarn’s Partial Inverse Method with Applications to Operator Splitting and Composite Optimization

In this paper, a pursuit-evasion game involving two non-holonomic agents is examined using the theory of differential games. It is assumed that the two players move on the Euclidean plane with fixed but different speeds and they each have a lower bound on their achievable turn radii. Both players steer at each instant by choosing their turn radii value and directions of turn. By formulating the game as a game of kind, we characterize the regions of initial conditions that lead to capture as well as the regions that lead to evasion, when both the players play optimally. The game is then formulated as a game of degree to obtain time-optimal paths for the pursuer and evader inside a capture region. Besides, all possible scenarios are considered for both players that differ in speed ratios and maneuverability constraints. Solutions are provided for those cases using appropriate simulation parameters, which aid in understanding the characteristics of the game of two cars under a wide range of constraints.     

Robust pursuit of a hybrid evader

A pursuit-evasion differential game with bounded controls and prescribed duration is considered. The evader has two possible dynamics, while the pursuer dynamics is fixed. The evader can change the dynamics once during the game. The pursuer knows the possible dynamics of the evader, but not the actual one. The optimal pursuer strategy in this game is obtained. It is robust with respect to the control of the evader, the order of its dynamics and the time of the mode change. The capture conditions of the game are established and the pursuer capture zone is constructed. An illustrative example of the game is also presented.     

A Two-on-One Linear Pursuit–Evasion Game with Bounded Controls

A linearized engagement with two pursuers versus a single evader is considered, in which the adversaries’ controls are bounded and have first-order dynamics and the pursuers’ intercept times are equal. Wishing to formulate the engagement as a zero-sum differential game, a suitable cost function is proposed and validated, and the resulting optimization problem and its solution are presented. Construction and analysis of the game space is shown, and the players’ closed-form optimal controls are derived for the case of two “strong” pursuers. The results are compared to those of a 1-on-1 engagement with a “strong” pursuer, and it is shown that the addition of a second pursuer enlarges the capture zone and introduces a new singular zone to the game space, in which the pursuers can guarantee equal misses, regardless of the evader’s actions. Additionally, it is concluded that in the regular zones the closed-form optimal pursuit strategies are unchanged compared to two 1-on-1 engagements, whereas the optimal evasion strategy is more complex. Several simulations are performed, illustrating the adversaries’ behavior in different regions of the game space.     

Quasilinear conflict-controlled processes with additional restrictions

A class of conflict-controlled processes [1–3] with additional (“phase” type) restrictions on the state of the evader is considered. A similar unrestricted problem was considered in [4]. Unlike [5, 6] the boundary of the “phase” restrictions is not a “death line” for the evader. Sufficient conditions for the solvability of the pursuit and evasion problems are obtained, which complement a range of well-known results [5–10].^‡     

Informational Sets in a Model Problem of Homing

A linear pursuit problem in the plane under incomplete pursuer information about the evader is investigated. At discrete time instants, the pursuer measures with errors the angle of vision to the evader, the angular velocity of the line of sight, and the relative distance. Other combinations of measurable parameters are possible (for example, angle of vision and relative distance or angle of vision only). The measurements errors obey certain geometric constraints. The initial uncertainties on the evader coordinates and velocities are given in advance. Having a resource of impulse control, the pursuer tries to minimize the miss distance. The evader control is bounded in modulus.The problem is formulated as an auxiliary differential game. Here, the notion of informational set is central. The informational set is the totality of pointwise phase states consistent with the history of the observation-control process. The informational set depends on the current measurements; it changes in time and plays the role of a generalized state, which is used for constructing the pursuer control.A control method designed for the linear pursuit problem is used in the planar problem of a vehicle homing toward a dangerous space object. The nonlinear dynamics is described by the Kepler equations. Nonlinear terms of the equations in relative coordinates are small and are replaced by an uncertain vector parameter, which is bounded in modulus and is regarded as an evader control. As a result, we obtain the mentioned control problem in the plane.The final part of the paper is devoted to the simulation of a space vehicle homing toward a dangerous space object. In testing the control method developed, two variants are considered: random measurement errors and game method of constructing the measurements; the latter is also described in the paper.     

Optimal Evasion from a Pursuer with Delayed Information

A class of prescribed duration pursuit–evasion problems with first-order acceleration dynamics and bounded controls is considered. In this class, the pursuer has delayed information on the lateral acceleration of the evader, but knows perfectly the other state variables. Moreover, the pursuer applies a strategy derived from the perfect information pursuit–evasion game solution. Assuming that the evader has perfect information on all the state variables as well as on the delay of the pursuer and its strategy, an optimal evasion problem is formulated. The necessary optimality conditions indicate that the evader optimal control has a bang–bang structure. Based on this result, two particular cases of the pursuer strategy (continuous and piecewise continuous in the state variables) are considered for the solution of the optimal evasion problem. In the case of the continuous pursuer strategy, the switch point of the optimal control can be obtained as a root of the switch function. However, in the case of the piecewise continuous (bang–bang) pursuer strategy, this method fails, because of the discontinuity of the switch function at this very point. In this case, a direct method for obtaining the switch point, based on the structure of the solution, is proposed. Numerical results illustrating the theoretical analysis are presented leading to a comparison of the two cases.     

Point capture of two evaders in succession

A constant-speed coplanar model with unlimited turn-rates leads to a rather simple geometrical solution of the problem of point capture of two successive evaders in minimum total time: the pursuer concentrates first on the nearer evader who runs in an appropriate direction; the second evader runs directly away from the predictable point of capture of the first evader. This simple solution is valid only if the second evader remains thereby the further of the two evaders. Otherwise, the solution must be modified to include a phase involving curved motion by all three players, during which the pursuer remains equidistant from both evaders.     

Stochastic pursuit-evasion differential games in 3D

A stochastic version of Isaacs's homicidal chauffeur game in the (x, y, z)-space is considered. This is used to solve a pursuit-evasion problem in the (x, y, z)-space in which the pursuer has incomplete information on the evader motion. Optimal feedback strategies for the game, and optimal feedback guidance laws for the pursuer, which uses only the measurements available to the pursuer, are computed. A simple suboptimal guidance law for the pursuer is suggested.     

Application of the epsilon technique to a realistic optimal pursuit-evasion problem

The Balakrishnan epsilon technique is applied to a pursuit—evasion problem in which both the pursuer and evader act optimally. The problem consists of determining the control functions for both the pursuer and the evader that result in the time of interception being minimized and maximized, respectively. The pursuer is considered to be a point mass confined to a horizontal plane and subjected to realistic aerodynamic and propulsive forces which are nonlinear functions of the state and the control. The evader has a constant speed and has a limit on the rate with which it can change direction. The Balakrishnan epsilon technique involves the insertion of a penalty term for not satisfying the dynamic equations directly. A modified Newton-Raphson method is used to solve the coefficients of a functional expansion of the state variables which minimizes the cost; a simple search is used to find the control at each time point which minimizes the integrand of the cost function. Repeated application of both the Newton-Raphson method and the search results in rapid convergence to the minimum-time solution to a fixed point. Six to eight iterations are typically required. A complete family of minimum-time flight paths to points in several directions are computed and subsequently used to determine the intercept point for virtually any evader flight path. Interpolation between solutions yields both the optimal path and corresponding control for the interception. A similar set of solutions are generated for the evader. The general solution of the evader is superimposed on that of the pursuer in a way that is consistent with the problem's initial conditions. The intercept point for the max-min solution corresponds to the maximum of the minimum time required for both the pursuer and evader to reach that point. Once the intercept point is determined, the corresponding trajectories and control functions for both the pursuer and evader can be obtained by interpolating between adjacent solutions. The approach used is particularly efficient when a large number of optimal pursuit-evasion solutions are needed.     

Simple-motion pursuit-evasion differential games,part 1: Stroboscopic strategies in collision-course guidance and proportional navigation

We consider pursuit-evasion differential games in the plane in which the players, i.e., the pursuer and the evader, have simple motion and are pedestrians à la Isaacs. Two information patterns are considered, namely the classical feedback strategy and the stroboscopic pursuit strategy; loosely speaking, the latter incorporates the instantaneous control employed by the evader, which we assume to be known to the pursuer. Within this framework, the question of modelling a pursuit-evasion encounter is addressed, and we examine three well-known guidance schemes of the line-of-sight, collision course, and proportional navigation types.     

The optimal pursuit problem reduced to an infinite system of differential equations

The optimal game problem reduced to an infinite system of differential equations with integral constraints on the players’ controls is considered. The goal of the pursuer is to bring the system into the zeroth state, while the evader strives to prevent this. It is shown that Krasovskii's alternative is realized: the space of states is divided into two parts so that if the initial state lies in one part, completion of the pursuit is possible, and if it lies in the other part, evasion is possible. Constructive schemes for devising the optimal strategies of the players are proposed, and an explicit formula for the optimal pursuit time is derived.     

Simple-motion pursuit-evasion differential games,part 2: Optimal evasion from proportional navigation guidance in the deterministic and stochastic cases

The problem of the optimal evasion from a pursuer employing proportional navigation guidance is considered. Thus, we consider the problem of maximizing the time-to-capture when the problem parameters, namely, the speed ratio and the proportional navigation constant, are such that capturability of the evader is guaranteed.     

A multistage pursuit-evasion game that admits a gaussian random process as a maximin control policy

In this paper we consider the existence and structure of both minimax and maximin policies for the special class of LQG pursuit-evasion games which is characterized by (i) a blind evader; and (ii) a pursuer who can make use of noise corrupted state measurements. The particular class of games which we consider has been studied previously by other investigators who have shown that pure strategies exist for both players. The major contribution of our paper is the delineation of the existence and structure of a mixed strategy for the evader in this class of games. This new maximin strategy is defined by a gaussian measure, which can be determined explicitly by the method of least favorable prior distributions. We show that the validity of the pure solutions determined previously is limited by the duration of the game, due to the existence of a ‘pure solution conjugate point’; further, we prove that our new strategies are valid solutions which extend the possible duration of the game beyond the limit imposed by the pure solution conjugate point. We believe that our paper constitutes the first report on the existence of a mixed strategy for an LQG game, and the first report on the role conjugate points play in the transition between pure strategies and mixed strategies.