A planning problem combining optimal control and optimal transport

We consider some variants of the classical optimal transport where not only one optimizes over couplings between some variables x and y but also over some control variables governing the (possibly stochastic) evolutions of these variables with time. Such a situation is motivated by an assignment problem of tasks with workers whose characteristics can evolve with time (and be controlled). We distinguish between the coupled and decoupled case. The coupled case is a standard optimal transport with the value of some optimal control problem as cost. The decoupled case is more involved since it is nonlinear in the transport plan. We fully treat the linear quadratic case and show that even in this case existence may fail for the decoupled problem.