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This article is devoted to a survey of results for differential games obtained through Viability Theory. We recall the basic theory for differential games (obtained in the 1990s), but we also give an overview of recent advances in the following areas: games with hard constraints, stochastic differential games, and hybrid differential games.We also discuss several applications.

This chapter deals with the pursuit games in which players (pursuer, evader, or both) employ impulse control.We consider continuous-time dynamical systems modeled by ordinary differential equations that are affected by jumps in state at discrete instants. The moments of jump comply with the condition for a finite number of jumps in finite time. In so doing, the Dirac delta function is used to describe the impulse control. Such systems represent a special case of hybrid systems. The method of resolving functions provides a general framework for analysis of the above-mentioned problems. This method essentially uses the technique of the theory of set-valued mappings. The following cases are examined in succession: impulse control of the pursuer; impulse control of the evader; impulse control of both players. The problem of approaching a cylindrical terminal set is studied for each case, and the sufficient conditions for its solvability are derived. Obtained results are supported by a model example of the game with simple motion dynamics.

For a simple nonzero-sum differential game on the real line, the natural notion of Nash equilibrium payoff in feedback form turns out to be extremely unstable. Two examples of different types of instability are discussed.

An approach for constructing a robust feedback control for problems having linear dynamics with disturbances is suggested.Auseful control is assumed to be scalar and bounded. The approach can be applied to conflict-controlled systems, where the constraint for the disturbance is unknown in advance. Adjustment of the method is based on results of the theory of linear differential games with fixed terminal time and geometric constraints for the players’ controls. The algorithm for constructing the robust control is fulfilled as a computer program for the case when the quality of the process is defined only by two components of the phase vector at the terminal instant. The paper presents simulation results for the problem of lateral motion control of an aircraft during landing under wind disturbance.

In classical noncooperative matrix games the payoffs are determined directly by the players’ choice of strategies. In reality, however, a player may not be capable of executing his or her chosen strategy due to a lack of skill that we shall refer to as incompetence. A method for analysing incompetence in matrix games is introduced, examined and demonstrated. Along with the derivation of general characteristics, a number of interesting special behaviours are identified. These special behaviours are shown to be the result of particular forms of the game and/or of the incompetence matrices. The development of this simple model was motivated by applications where a decision to increase competence is to be evaluated. Investments that decrease incompetence (such as training) need to be related to the change in the value of the game. A game theory approach is used since possible changes in strategies used by other players must be considered. An analogy with the game of tennis is discussed, as is an analogy with capability investment decisions in the military.

We consider the Stackelberg well-posedness for hierarchical potential games and relate it to the Tikhonov well-posedness of the potential function as a maximum problem. We also make some considerations about the strong and weak Stackelberg approximate equilibria.

We present and study a notion of ergodicity for deterministic zero-sum differential games that extends the one in classical ergodic control theory to systems with two conflicting controllers.We show its connections with the existence of a constant and uniform long-time limit of the value function of finite horizon games, and characterize this property in terms of Hamilton-Jacobi-Isaacs equations.We also give several sufficient conditions for ergodicity and describe some extensions of the theory to stochastic differential games.

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that the extension of the solution policy to a later starting time and any possible state brought about by prior optimal behavior of the players will remain optimal. Recently, mechanisms for the derivation of subgame consistent solutions in stochastic cooperative differential games with transferable payoffs have been found. In the case when players’ payoffs are nontransferable, the derivation of solution candidates is extremely complicated and often intractable. In this chapter, subgame consistent solutions are derived for a class of cooperative stochastic differential games with nontransferable payoffs.

Pursuit-evasion games with simple motion on two-dimensional (2D) manifolds are considered. The analysis embraces the game spaces such as 2D surfaces of revolution (cones and hyperboloids of one and two sheets, ellipsoids), a Euclidean plane with convex bounded obstacle, two-sided Euclidean plane with hole(s), and two-sided plane bounded figures (disc, ellipse, polygon). In a two-sided game space players can change the side at the boundary (through the hole). In all cases the game space is a 2D surface or a figure in 3D Euclidean space, while the arc length is induced by the Euclidean metric of the 3D space. Due to simple motion, optimal trajectories of the players generally consist of geodesic lines of the game space manifolds. For the game spaces under consideration there may exist two or more geodesic lines with equal lengths, connecting the players. In some cases this gives rise to a singular surface consisting of the trajectories, which are envelopes of a family of geodesics. In this chapter we investigate the necessary and sufficient conditions for this and some other types of singularity, we specify the game spaces where the optimal pursuit-evasion strategies do not contain singularities and are similar to the case of a Euclidean plane, and we give a short review and analysis of the solutions for the games in several game spaces-manifolds. In the analysis we use viscosity solutions to the Hamilton-Jacobi-Bellman-Isaacs equation, variation calculus and geometrical methods.We also construct the 3D manifolds in the game phase space representing the positions with two or more geodesic lines with equal lengths, connecting the players. The investigation of 2D games on the manifolds has several direct applications, and it may also represent an approximate solution for more complicated games as an abstraction.

A class of pursuit-evasion differential games with bounded controls and a prescribed duration is considered. Two finite sets of possible dynamics of the pursuer and evader, known for both players, are given. The evader chooses his dynamics once before the game starts. This choice is unavailable for the pursuer, which causes a dynamics uncertainty. The pursuer can change his dynamics a finite number of times during the game, yielding a variable structure dynamics. The solution of this game is derived including optimal strategies of the players. The existence of a saddle point is shown. The game value and the shape of the maximal capture zone are obtained. Illustrative examples are presented.