Link between controllability of ODEs and controllability of transport equations What is the relationship between the controllability of the ODE
$$\dot x(t) = v(x) + u(t)$$
using a control $u$ and the controllabilty of the transport equation
$$\rho_t(t,x) + \mathrm{div}(v(x) \rho(t,x) + \mathbf{1}_\omega u(t)) = 0$$
where $\omega \subset \mathbb R^n$ with a control $\mathbf{1}_\omega u(t)$?
Does one imply the other? Are they equivalent? Where can I find references of the controllability of either (or both) of these problems?
 A: While there exist a couple of established links between the controllability properties of continuity equations and ODEs, there is not a unique and straightforward answer to your question.
Indeed, while controllability is a rather univocal notion at the ODE level, what can be said at the level of the PDE depends on extra considerations, such as the regularity imposed on the velocity field $v(\cdot)$, the class of solutions that are considered (general measures, functions...), the notion of controllability that one seeks to achieve (exact, approximate...), and possibly on some geometric condition imposed on the domain $\omega$.
To the best of my knowledge, there exists two (fairly recent) papers dealing with this topic by my former PhD advisor and two of his co-authors.

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*https://epubs.siam.org/doi/abs/10.1137/17M1152917

*https://www.sciencedirect.com/science/article/pii/S0022039619306321
If you are interested in general control-theoretic references on the notion of controllability, you should probably look into the classical Chow-Rashevskii theorem that is nicely presented in https://hal.archives-ouvertes.fr/hal-01137580/file/F-Jean_Nonholonomic.pdf
A: Here's another references:Elamvazhuthi, Karthik, et al. "Bilinear controllability of a class of advection–diffusion–reaction systems." IEEE Transactions on Automatic Control 64.6 (2018): 2282-2297.
