In Section 6.2 we used dynamic programming to derive the nonlinear partial differential equation (12.1.1) for the value function associated with an optimal control problem. This partial differential equation is called a HamiltonJacobi-Bellman (HJB) equation, also Bellman’s equation. Typically, the value function W is not smooth, and (12.1.1) must be understood to hold in some weaker sense. In particular, under suitable assumptionsW satisfies (12.1.1) in the Crandall-Lions viscosity solution sense (Section 12.5). Section 12.6 gives an alternate characterization (12.6.2) of the value function using lower Dini derivatives. This provides a control theoretic proof of uniqueness of viscosity solutions to the HJB equation with given boundary conditions.