Control of diffeomorphisms and densities

Consider a classical control system as it was deЇned by Pontryagin:
\begin{equation} \dot x=f(x,u), \qquad x\in M, \quad u\in U, \tag{1} \end{equation}
Assume that the state space $M$ is a smooth manifold, the set of control parameters $U$ is a closed subset of another smooth manifold, the right-hand side $f$ is smooth, and a reasonable completeness assumption allows to extend solutions of ordinary differential equations to the whole time axis.
We call controls the mappings $\mathbf u\colon(t,x)\mapsto\mathbf u(t,x)$ with values in $U$ that are smooth with respect to $x$ and measurable bounded with respect to $t$: a mixture of the program and feedback controls. Now plug-in a control in system (1) and obtain a time-varying ordinary differential equation
\begin{equation} \dot x=f(x,\mathbf u(t,x)), \tag{2} \end{equation}
which generates a family of diffeomorphisms $P_t\colon M\to M$, where $P_0(x)=x$ and the curves $t\mapsto P_t(x)$ satisfy (2) for any $x\in M$. We say that $t\mapsto P_t$ is an admissible “trajectory” in the group of diffeomorphisms associated to the control $\mathbf u$.
Given an integral cost functional
$$ J(u(\,\cdot\,))=\int_0^T\varphi(x(t),u(t))\,dt $$
and a probability measure $\mu$ on $M$, we set
$$ \mathbf J_\mu(\mathbf u)=\int_0^T\int_M\varphi(P_t(x),\mathbf u(t,x))\,d\mu\,dt $$
a functional on the space of controls $\mathbf u$.
In my talk, I am going to discuss the controllabilty and optimal control issues for the defined in this way systems on the group of diffeomorphisms.