We consider the SDE (just assume we are in $\mathbb{R}$, i.e. $X_t\in\mathbb{R}$): \begin{equation*} dX_t=v(X_t,t)dt+dB_t \quad t\in[0,1] \end{equation*} Here $v(\cdot,\cdot)$ is our control, my question is: Given the distribution of $X_0$ and $X_1$, suppose $X_0\sim \mu_0$ and $X_1\sim \mu_1$.
Does there exist such control $v(\cdot,\cdot)$ that satisfy the boundary condition? If so, what is the constraint on $v(\cdot,\cdot)$?
(I know that the solution to the so called Schrodinger Problem will give a solution to such control, so the answer to the first question is probably Yes, I am mainly interested in the second question.)