TABLE OF CONTENTS This chapter deals with two-person deterministic zero-sum differential games with variable terminal time. We have already discussed zero-sum differential games in Chapter 6, but as a special case of nonzero-sum games and with fixed terminal time. The class of problems to be treated in this chapter, however, are of the pursuit-evasion type for which the duration of the game is not fixed. Actually, it was through this type of problems (i.e., through the study of pursuit and evasion between two objects moving according to simple kinematic laws) that the theory of differential games was started in the early 1950s. Extensions to nonzero-sum dynamic games, as treated in Chapters 6 and 7, were subsequently considered in the late 1960s.
Section 8.2 discusses the necessary and sufficient conditions for existence of saddle-point equilibrium strategies. Sufficiency conditions are provided by a natural two-person extension of the Hamilton Jacobi Bellman equation, which is called the Isaacs equation , after Isaacs the acknowledged father of pursuitevasion games. A geometric derivation of the Isaacs equation is given, which utilizes the principle of dynamic programming, and by means of which the concept of semipermeable surfaces is introduced; such surfaces play an important role in the remaining sections of the chapter. Subsequently, necessary conditions are derived, which form the two-person extension of the Pontryagin minimum principle (cf. Section 5.5). These conditions are valid not for feedback strategies but for their open-loop representations, and in order to obtain the feedback strategies these open-loop solutions have to be synthesised. We also briefly...
