There is some point $Y$, which follows the distribution $p_{Y}(y)$ on $\mathbb{R}$, which we will assume to be normally distributed with $\sigma^2=1$ and $\mu = 0$. We wish to find some path $x(t)$, with $x(0)=0$, which minimizes \begin{equation} \mathbb{E}_{Y}[t_0(y, x(t))], \end{equation} where $t_0$ is the least amount of time it takes to reach $y$ by taking the path $x(t)$, under some velocity constraint $|x'(t)| \leq 1$. It seems like this problem should be solvable via a variational technique. We can write the expectation as some functional of $x(t)$: \begin{equation} S[x(t)] = \int_{\mathbb{R}}x^{-1}(y)p_{Y}(y)dy, \end{equation} where $x^{-1}(y) = \min \left(\left\{t : x(t) = y\right\}\right)$. I think the optimal solution $x(t)$, must not be one-to-one, so it is necessary to define some inverse in this way.
I've been playing around with different formulations, trying to compute $\frac{\delta S}{\delta x}$, but I have not had much luck. Does anyone have an insight into this problem? How might I compute the minimizing $x(t)$?