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.     

Homicidal Chauffeur Game with Target Set in the Shape of a Circular Angular Sector: Conditions for Existence of a Closed Barrier

A planar constant-speed pursuit-evasion problem with dynamic model similar to the one of the homicidal chauffeur game and with prescribed angular constraints in the capture criterion is analyzed as a differential game of kind. Because of the angular constraints, the target set of the game has the shape of a circular angular sector. Conditions for the existence of the game barrier (closed) are obtained. Using these conditions, a necessary and sufficient condition for capture of a slower evader from any initial state of the game is established. This condition is represented by an expression for the minimal nondimensional capture radius, normalized by the pursuer minimal turning radius, which guarantees capture of all slower evaders. This minimal capture radius depends on the angular constraints. Capture from any initial state implies that the barrier of the game does not exist and vice versa. In this game, two types of barrier are derived, with termination at either points of smoothness or points of nonsmoothness (corner points) of the boundary of the target set. The results are illustrated by numerical examples.     

The bang-bang-bang problem revisited

The solution given by Isaacs to the bang-bang-bang problem is shown to be a solution in aweak sense only. A solution in a stronger sense is demonstrated involving the notions of extended closed-loop strategies and of extended value-function. The bang-bang-bang singular surface is interpreted in a new way.The author is grateful to Professor J. V. Breakwell for his criticism and valuable remarks.     

Two-dimensional pursuit-evasion game with penalty on turning rates

An approach, both analytical and numerical, is used to solve a two-dimensional pursuit-evasion game characterized by a difficulty level intermediate between that of thesimple motion game (with freely and instantaneously oriented velocities) and that of thegame of two cars (with lower bounds on curvature radii). Each player's velocity has a constant modulus. The maneuvers are penalized by introducing, in the performance index, an integral term for the squared velocity turning rate.The local problem solution is relatively easy to find: the equations of motion and the adjoint equations can be integrated by means of elliptic functions and integrals. The global problem is more delicate to solve, because of the existence of a dispersal singular surface requiring an important numerical search to be determined. Thesynthesis problem (how to express the optimal strategies as functions of state) is not explicitly solvable, but a numerical approach using successive approximations can be developed. Illustrative interception trajectories are given.The authors are grateful to Mr. J. P. Peltier, Head, Guidance Group, Aerospace Mechanics Division, Systems Department, ONERA, Châtillon, France, for his suggestions and his efficient assistance in the numerical aspect of this study.     

Conic surveillance evasion

A surveillance-evasion differential game of degree with a detection zone in the shape of a two-dimensional cone is posed. The nature of the optimal strategies and the singular phenomena of the value function are described and correlated to subsets of the space of all possible parameter combinations, showing the relation of the singular phenomena in differential game theory and control theory.The first author was partially supported by the Stichting Hogeschoolfonds Twente, Enschede, The Netherlands.     

Games of Simple Pursuit and Approach on a Two-Dimensional Cone

The complete solution of two differential games with simple motion on a two-dimensional cone is presented. The method of generalized (singular) characteristics is used for the analysis of singular paths. The level lines of the game value and optimal phase portrait are constructed, and the latter is shown to include several types of singular surfaces (dispersal, equivocal, universal). An algorithm is developed for the construction of the synthesis for both players and is used in computer simulation.     

Characterization of Barriers of Differential Games

In pursuit-evasion games, when a barrier occurs, splitting the state space into capture and evasion areas, in order to characterize this manifold, the study of the minimum time function requires discontinuous generalized solutions of the Isaacs equation. Thanks to the minimal oriented distance from the target, we obtain a characterization by approximation with continuous functions. The barrier is characterized by the largest upper semicontinuous viscosity subsolution of a variational inequality. This result extends the Isaacs semipermeability property.     

Simple Singularities for Hamilton-Jacobi Equations with Max-Concave Hamiltonians and Generalized Characteristics

We consider piecewise smooth viscosity solutions to a Hamilton-Jacobi equation, for which the Hamiltonian is the maximum of a finite number of smooth concave functions. We describe the possible types of “basic” singularities, namely jump discontinuities in either the derivative of the solution or the Hamiltonian, which occur across a smooth hypersurface. Each such type of singularity is described in terms of the properties of the Hamiltonian and the classical characteristics in the regions on either side of the singularity. Where appropriate, singular characteristic equations of the types developed by Melikyan are formulated which can be used in constructions.     

Complex differential games of pursuit-evasion type with state constraints,part 1: Necessary conditions for optimal open-loop strategies

Complex pursuit-evasion games with state variable inequality constraints are investigated. Necessary conditions of the first and the second order for optimal trajectories are developed, which enable the calculation of optimal open-loop strategies. The necessary conditions on singular surfaces induced by state constraints and non-smooth data are discussed in detail. These conditions lead to multi-point boundary-value problems which can be solved very efficiently and very accurately by the multiple shooting method. A realistically modelled pursuit-evasion problem for one air-to-air missile versus one high performance aircraft in a vertical plane serves as an example. For this pursuit-evasion game, the barrier surface is investigated, which determines the firing range of the missile. The numerical method for solving this problem and extensive numerical results will be presented and discussed in Part 2 of this paper; see Ref. 1.This paper is dedicated to the memory of Professor John V. Breakwell.The authors would like to express their sincere and grateful appreciation to Professors R. Bulirsch and K. H. Well for their encouraging interest in this work.     

Singular surfaces in a linear pursuit-evasion game with elliptical vectograms

In a linear pursuit-evasion game with elliptical vectograms, singular surfaces appear in the plane of symmetry containing the ellipses minor axes and, as a consequence, locally nonsmooth isocost tubes are generated. Both dispersal and attractive focal surfaces are encountered. The dispersal surface is a zone of initial conditions for trajectories leaving the plane of symmetry. Optimal trajectories attracted by the focal surface merge tangentially to the plane of symmetry and remain there until the boundary of the dispersal zone is reached. Determination of the saddle-point strategies in the focal surface leads to constructing the isocost surfaces in the entire game space.     

The partition bargaining problem

Consider the problem of partitioning n items among d players where the utility of each player for bundles of items is additive; so, player r has utility for item i and the utility of that player for a bundle of items is the sum of the 's over the items i in his/her bundle. Each partition S of the items is then associated with a d-dimensional utility vector V^S whose coordinates are the utilities that the players assign to the bundles they get under S. Also, lotteries over partitions are associated with the corresponding expected utility vectors. We model the problem as a Nash bargaining game over the set of lotteries over partitions and provide methods for computing the corresponding Nash solution, to prescribed accuracy, with effort that is polynomial in n. In particular, we show that points in the pareto-optimal set of the corresponding bargaining set correspond to lotteries over partitions under which each item, with the possible exception of at most d(d-1)/2 items, is assigned in the same way.     

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.     

Feedback Nash equilibria in a problem of optimal fishery management

The paper deals with a problem of optimal management of a common-property fishery, modelled as a two-player differential game. Under nonclassical assumptions on harvest rates and utilities, a feedback Nash equilibrium is determined, using a bionomic equilibrium concept. Later on, this assumption is relaxed and a feedback Nash equilibrium is established under minimal hypotheses.     

Zero-sum state constrained differential games: existence of value for Bolza problem

We prove the existence of a lower semicontinuous value function for Bolza problem in differential games with state-constraints. As a byproduct, we obtain a new estimation of trajectories of a control system by trajectories with state constraints. This result which could be interesting by itself enables us to build a suitable strategy for constrained differential games. We also characterize the value function by means of viscosity solutions and give conditions under which the value function is locally Lipschitz continuous.Work supported by the European Community’s Human Potential Program under contract HPRN-CT-2002-00281, [Evolution Equations].     

A double obstacle problem arising in differential game theory

A Hamilton–Jacobi equation involving a double obstacle problem is investigated. The link between this equation and the notion of dual solutions—introduced in [S. As Soulaimani, Infinite horizon differential games with asymmetric information, PhD thesis; P. Cardaliaguet, Differential games with asymmetric information, SIAM J. Control Optim. 46 (3) (2007) 816–838; P. Cardaliaguet, C. Rainer, Stochastic differential games with asymmetric information, Appl. Math. Optim. 59 (1) (2009) 1–36] in the framework of differential games with lack of information—is established. As an application we characterize the convex hull of a function in the simplex as the unique solution of some nonlinear obstacle problem.     

Qualitative analysis of secured outcome regions for two-target games

Qualitative (game of kind) outcomes of two-target games are analyzed in this paper, under both the zero-sum and nonzero-sum preference ordering of outcomes by the players. The outcome regions of each player are defined from a security standpoint. The secured draw and mutual-kill regions of a player depend explicitly on his preference ordering of outcomes and should be constructed separately for each player, especially in a nonzero-sum game. General guidelines are presented for identifying the secured outcome regions of players in a class of two-target games that satisfy an Isaacs-like condition, in terms of the qualitative solutions of the two underlying single-target pursuit-evasion games. A construction has been proposed for obtaining the qualitative solution of a large class of two-target games. Illustrative examples are included.This work was done while the first author was a Research Associate in the Department of Electrical Engineering at the Indian Institute of Science, Bangalore, and was financially supported by the Council of Scientific and Industrial Research, Delhi, India.     

The Leitmann-Schmitendorf advertising game with n players and time discounting

The extension of the Leitmann-Schmitendorf advertising game to n players and positive time discounting is investigated. We show that the strong time consistency of the open-loop Nash equilibrium is preserved. As to optimal controls, while the boundary solution is unaffected by the number of firms as well as discounting, the inner solution depends on industry structure. The fully symmetric version of the game allows us to identify the parameter regions wherein both solutions are sustainable.     

The affine Cauchy problem

The aim of this paper is to solve the Cauchy problem for locally strongly convex surfaces which are extremal for the equiaffine area functional. These surfaces are called affine maximal surfaces and here, we give a new complex representation which let us describe the solution to the corresponding Cauchy problem. As applications, we obtain a generalized symmetry principle, characterize when a curve in R³ can be a geodesic or pre-geodesic of a such surface and study the helicoidal affine maximal surfaces. Finally, we investigate the existence and uniqueness of affine maximal surfaces with a given analytic curve in its singular set.