All papers I have read define the Hamilton-Jacobi-Bellman equation starting from the Bolza Problem like so:
$$\min_u J=\phi(x(t_f))+\int_{t_0}^{t_f} L(x(t),u(t),t)dt$$
subjected to
$$ \frac{dx(t)}{dt} = f(x(t),u(t),t), \quad x(t_0)=x_0. $$
The value function can then be defined as
$$ V(t,x(t))=\min_u \int_t^{t_f} L(x(\tau),u(\tau),\tau)d\tau + \phi(x(t_f)).$$
The HJB is then
$$ -\frac{\partial V(t,x)}{\partial t} = \min_u (L(x,u,t) + \frac{\partial V(t,x)}{\partial x} f(x,u,t))$$
where
$$ V(t_f,x(t_f)) = \phi(x(t_f)).$$
My question is what is the HJB when $L(x,u,t)= 0$ (i.e., the Mayer problem)? It seems obvious that the HJB should be
$$ 0 = \min_u (\frac{\partial V(t,x)}{\partial x} f(x,u,t)).$$
$\frac{\partial V(t,x)}{\partial t}$ is set to zero because $V(t,x(t))=\phi(x(t_f))$ which is time independent. For the Mayer problem, $\frac{\partial V(t,x)}{\partial x} f(x,u,t)$ is effectively the Hamiltonian, which is only equal to 0 throughout the time domain if the final time is optimal. Thus, if the predefined $t_f$ is not optimal, then the HJB I proposed for the Mayer problem should be incorrect (I believe)? Can anyone define the HJB for the Mayer problem, or point me in the direction to a resource that does?