I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max_{v \in C^1([a, b])}   \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \leq 0,~ \frac{o(b)}{o(a)} \leq v \leq 1
$$
where $o \in C^\infty(\mathbb{R})$ with $o>0$ and $o'< 0$ on $(a, b)$ and $f \in C([a, b])$ with $f>0$ on $(a, b)$. I would already be very happy if someone could provide me with input for the case $f(x)=1$. 

Problem is, I can find little to no publications on such problems. Moreover, the usual optimality conditions are useless because we lose $v$ in the derivative of both objective and constraint. My suspicion is that
$$
v(x) = \frac{o(b)}{o(x)}
$$
for all $x \in [a, b]$ is optimal, since it solves the ODE $-o'v-v'o$ and is $1$ on the boundary $b$. 

Furthermore, using Gronwall, I was able to show that if some optimal $\tilde{v}$ exists and
$$
-o'(x)\tilde{v}(x) -\tilde{v}'(x)o(x) \leq 0
$$
on some interval $(c, d) \subseteq (a, b)$ holds, then we get the upper bound
$$
\tilde{v}(x) \leq \frac{o(d)}{o(x)}
$$
on $\tilde{v}$ for all $x \in (c, d)$, which is at least somewhat consistent with my hypothesis. Any input would really be appreciated.