Differential games of fixed duration with state constraints

We consider differential games of fixed duration with phase coordinate restrictions on the players. Results of Ref. 1 on games with phase restrictions on only one of the players are extended. Using Berkovitz's definition of a game (Ref. 2), we prove the existence and continuity (or Lipschitz continuity) of the value under appropriate assumptions. We also note that the value can be characterized as the viscosity solution of the associated Hamilton-Jacobi-Isaacs equation.This work comprises a part of the author's PhD Thesis completed at Purdue University under the direction of Professor L. D. Berkovitz. The author wishes to thank Professor Berkovitz for suggesting the problem and many valuable discussions. During the research for this work, the author was supported by a David Ross Grant from Purdue University as well as by NSF Grant No. DMS-87-00813.     

Repeated games with public uncertain duration process

We consider repeated games where the number of repetitions θ is unknown. The information about the uncertain duration can change during the play of the game. This is described by an uncertain duration process Θ that defines the probability law of the signals that players receive at each stage about the duration. To each repeated game Γ and uncertain duration process Θ is associated the Θ-repeated game Γ_Θ. A public uncertain duration process is one where the uncertainty about the duration is the same for all players. We establish a recursive formula for the value V _Θ of a repeated two-person zero-sum game Γ_Θ with a public uncertain duration process Θ. We study asymptotic properties of the normalized value v _Θ = V _Θ/E(θ) as the expected duration E (θ) goes to infinity. We extend and unify several asymptotic results on the existence of lim v _n and lim v _λ and their equality to lim v _Θ. This analysis applies in particular to stochastic games and repeated games of incomplete information.     

Differential games with random noise

A differential game is considered in which an opponent is a random noise. An ally uses ε-strategies, defined in the classical theory of differential games. The mathematical expectation of possible harm caused by the opponent's actions is minimized. Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 78–89.     

Differential games with noise-corrupted measurements

In this paper, readily computable strategies for zero-sum, linear-quadratic differential games with noise-corrupted measurements are developed. Of particular significance is the fact that the governing differential equations no longer require the solution of an often difficult nonlinear, two-point boundary-value problem, but again satisfy the separation principle of linear-quadratic optimal control. The implications of the payoff relationships are considered.In a subsequent paper, we will apply the theory developed in this paper to a detailed example of a pursuit-evasion game. We discuss a missile and an airplane system where the missile supported by its launch platform has perfect state measurements and the airplane has noise-corrupted measurements.     

Differential games on partially overlapping sets

We suggest four new notions of optimality (equilibrium) and use them to construct a theory providing the existence and (almost always) the uniqueness of solutions of game problems (both static problems and problems described by differential equations) with partially overlapping game sets of the players; such problems, which are used when modeling conflict problems with incidental interests (or incidental profits) of the players, have not been studied in the classical theory.     

Differential inclusions and Young measures involving prescribed Jacobians

In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (e.g. for incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. This short note, which is based on joint work with K. Koumatos and E. Wiedemann, presents various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension. In particular, we give a characterization theorem for Young measures under this side constraint. This is in the spirit of the celebrated Kinderlehrer–Pedregal Theorem and based on convex integration and “geometry” in matrix space. Finally, applications to approximation in Sobolev spaces and to the failure of lower semicontinuity for certain integral functionals with “realistic” growth conditions are given. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Isaacs equation of differential games of fixed duration

The conditions under which the value function of fixed-duration differential games satisfies the Isaacs equation are relaxed.The author thanks Professor L. D. Berkovitz for posing the problem.     

Existence of value in differential games with fixed time duration

A differential game of prescribed duration with general-type phase constraints is investigated. The existence of a value in the Varaiya-Lin sense and an optimal strategy for one of the players is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. These results are next widened on more general games, examined earlier by Varaiya.The author wishes to express his thanks to an anonymous reviewer for his many valuable comments.     

Differential games without pure strategy saddle-point solutions

In some two-player, zero-sum differential games, pure strategy saddle-point solutions do not exist. For such games, the concept of a minmax strategy is examined, and sufficient conditions for a control to be a minmax control are presented. Both the open-loop and the closed-loop cases are considered.The research was partially supported by ONR under Contract No. N00014-69-A-0200-12. An earlier version of this paper was presented at the Eleventh Annual Allerton Conference on Circuit and System Theory, Monticello, Illinois, 1973.The author wishes to acknowledge his many valuable discussions of this problem with Professor G. Leitmann and also to thank one of the reviewers for his suggestions for simplifying the proof of Theorem 2.1.     

Differential games with a given value function

The set of solutions of a differential game with a terminal payoff functional is investigated. A method is obtained that allows us to establish whether a given function is a value of some differential game with a terminal payoff functional. The condition obtained is in fact the condition for the given function to be a minimax (viscosity) solution of some Hamilton-Jacobi equation with Hamiltonian homogeneous in the third variable. We also obtain a sufficient condition for a function to belong to the set of values of differential games with a terminal payoff function.     

Differential games with a given value function

For an integral equation describing stationary distributions of a biological community, we point out conditions on its parameters under which this equation has a unique solution that satisfies necessary requirements for such a distribution.     

Differential games with fixed terminal time and estimation of the instability degree of sets in these games

This paper contributes to the theory of differential games. A game problem of bringing a conflict-controlled system to a compact target set is analyzed. Sets in the position space that terminate on the target set and are not stable bridges are considered. The notion of stability defect of these sets is examined. It is demonstrated how the notion of stability defect can be used to construct sets with relatively good geometry that are at the same time convenient for the first player to play the game successfully.     

Differential games with information time lag: Norm-invariant systems

The theory of differential games with information lag is applied to an encounter between norm-invariant systems. The problem is solved in closed form to illustrate some of the effects of the information time lag.     

Differential games with continuous,switching and impulse controls

A two-person zero-sum differential game of infinite duration with discounted payoff involving hybrid controls is studied. The minimizing player is allowed to take continuous, switching, and impulse controls whereas the maximizing player is allowed to take continuous and switching controls. By taking strategies in the sense of Elliott–Kalton, we prove the existence of value and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities.