# Reference: Stochastic Optimal Control with cost functional

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 $$J$$).