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Maybe I am completely wrong here (given that I don't see the need to talk about viscosity solutions at all) but in the standard representation theorems you have a terminal/limit condition that the solution to the HJB has to satisfy for it to be the value function. This involves checking, for any admissible control,  \lim_{T \to \infty} E[ e^{-\rho T} V(a_T)...
Since you mention Walton, here is something from his notes page 111 Definition 4: The Hamiltonian and is defined $H(t,x,\rho) := min _v \{c(t,x,v) - \rho v\}$ Notice the analogue to the Hamiltonian you have written. $v$ in here is the control (your $c$), $c(t,x,v)$ in here is the $u(c)$ function you have and $-\rho = \partial_xL(t,x)$ in here where $L$ is ...