There are a variety of control problems for controlled diffusions $X_t^u$, with the terminal cost given by
$$
J(u)\triangleq \mathbb{E}\left[g(X_T,u)+\int_0^t h(X_t,u_t)ds\right],
$$
function $g$ and running cost $h$ are Lipchutz (and nice). Is there any literature on stochastic optimal control problems where the cost is given by
$$
\tilde{J}(u)\triangleq G(X_T,u) + \mathbb{E}\left[\int_0^t h(X_t,u_t)ds\right],
$$
where $G$ is a convex functional from $L^1(\mathcal{F}_T)\rightarrow \mathbb{R}$?
I'm looking for references and/or the name of this type of control problem. Thanks.
Note: (Of course the cost functionals of the form $\tilde{J}$ generalize those of the form $J$).
