Discontinuous viscosity solutions and applications

In this chapter we extend the theory of continuous viscosity solutions developed in Chapter II to include solutions that are not necessarily continuous. This has two motivations. The first is that many optimal control problems have a discontinuous value function and we want to extend to these problems the results of Chapters III and IV, in particular the characterization of the value function as the unique solution of the Hamilton-Jacobi-Bellman equation with suitable boundary conditions. The second motivation is more technical: viscosity solutions are stable with respect to certain relaxed semi-limits, that we call weak limits in the viscosity sense, which are semicontinuous sub- or supersolutions. These weak limits are used extensively in Chapters VI and VII to study the convergence of approximation schemes and several asymptotic limits, even for control problems where the value function is continuous.