A viscosity solutions

In this appendix we will briefly review the concept of viscosity solutions and discuss those properties that we need for treating the ISDS and wISDS Lyapunov functions. The theory of viscosity solutions started in the early 1980's with the papers by Crandall and Lions [24, 25], Crandall, Evans and Lions [23] and the monograph by Lions [89]. A comprehensive up-to-date overview (especially in connection with optimal control problems) can be found in the monograph by Bardi and Capuzzo Dolcetta [8]. Viscosity solutions provide a powerful solution concept for those partial differential equations (and also partial differential inequalities) which in general do not admit smooth classical solutions. In particular (but of course not exclusively) this applies to first order equations such as those appearing in the characterization of Lyapunov functions and in Zubov's method, both of which play a vital role in this book.