Problem: Given a controlled n-dimensional linear stochastic system on $[0,T]$, let's say:$$d\underline X(t)=A\underline X(t)dt + B\underline u(t)dt + d\underline W(t) $$ $$\ \underline X(0)=x \in \Bbb R^n$$ I would like to find a policy $\ \{\underline u(t)\}_{t \in [0,T]}$ that maximizes $\ \Bbb E[g(\underline X(T))]$ where g is a continuous function :$\ \Bbb R^n \rightarrow \Bbb R$. Now, the policy must satisfy also the constraints: $$\underline u(t)\geqslant \underline0 ,\forall t \in [0,T]$$ $$ \sum_1^n\int_0^Tu_i(s)ds=m \in \Bbb R$$ Question: is there a theory that deals with this problem? A very simple example: if I have a precise amount of money to give to a system of $\ N $ interconnected banks, how can I redistribute properly this money during time and among banks, for example, in order to maximize $\ \Bbb E[\sum_1^n\ X_i(T)]$?
I think that this would be classified as a stochastic control problem (find an open loop policy), but I never saw a constraint like this, and I've realized that also very simple constraint like the positiveness of the policy can really improve the difficulty of the classic problem. Is there maybe, another well studied approach to the problem of such a "dynamic allocation problem" different from stochastic control? Or maybe the stochastic control is the only standard way to solve optimal allocation in a dynamic stochastic context? I'm new to this kind of problems, I will appreciate any help.