An Introduction to Dynamic Programming and Viscosity Solutions

HJ equations have a long history, dating back at least to the calculus of variations of the 19th century, and HJ equations find wide application in science, engineering, etc. Perhaps surprisingly, it was only relatively recently that a satisfactory general notion of solutions for (1) became available, with the introduction of the concept of viscosity solution (Crandall-Lions, c. 1980). The difficulty, of course, is that solutions are not in general globally smooth (e.g. C). Solutions are often smooth in certain regions, in which the famous method of characteristics may be used to construct solutions. There are a number of other notions of solution available, such as encountered in non-smooth analysis (e.g. proximal solution), though we will not discuss them here.