The theory of two-person, zero-sum differential games started at the be-
ginning of the 1960s with the works of R. Isaacs in the United States
and L. S. Pontryagin and his school in the former Soviet Union. Isaacs
based his work on the Dynamic Programming method. He analyzed many
special cases of the partial differential equation now called Hamilton-
Jacobi-Isaacs-briefiy HJI-trying to solve them explicitly and synthe-
sizing optimal feedbacks from the solution. He began a study of singular
surfaces that was continued mainly by J. Breakwell and P. Bernhard and
led to the explicit solution of some low-dimensional but highly nontriv-
ial games; a recent survey of this theory can be found in the book by J.
Lewin entitled Differential Games (Springer, 1994). Since the early
stages of the theory, several authors worked on making the notion of
value of a differential game precise and providing a rigorous derivation
of the HJI equation, which does not have a classical solution in most
cases; we mention here the works of W. Fleming, A. Friedman (see his
book, Differential Games, Wiley, 1971), P. P. Varaiya, E. Roxin, R. J.
Elliott and N. J. Kalton, N. N. Krasovskii, and A. I. Subbotin (see
their book Po- sitional Differential Games, Nauka, 1974, and Springer,
1988), and L. D. Berkovitz. A major breakthrough was the introduction in
the 1980s of two new notions of generalized solution for Hamilton-Jacobi
equations, namely, viscosity solutions, by M. G. Crandall and P. -L.
