7.2 HJB equation, canonical equations, and characteristics

We know from Section 5.1.3 that the HJB equation is a PDE and is difficult to solve in general. In Section 5.2 we compared the HJB equation with the necessary conditions of the maximum principle and demonstrated, in particular, how solving the HJB equation for the value function $V$ formally leads to a solution of the canonical equations (via $p:=-{V}_{x}$ ). In this section we discuss an important piece of the PDE theory which enables one, in principle, to solve a PDE with the help of a suitable system of ODEs called the characteristics. We will show that the canonical equations arise as the characteristics of the HJB equation. Thus, compared to Section 5.2, the method of characteristics takes us in the opposite direction. We will first introduce this method in the context of general PDEs and then specialize it to the setting of the HJB equation.