Stochastic and Differential Games

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.