How to optimize a nested functional?

Question

I have to find the function $q(t)$ that optimizes the expected value of the integral $\int{V(q(t),t)dq}$ where t is a randomly distributed value with probability density $f(t)$. Thus, the functional to optimize would be $\int{f\int{Vq'dt}dt}$ My textbook just integrates by parts with the constant term for $F$ being $-1$ without any further explanation, though it makes sense to me why it should be either $0$ or $-1$. I've tried using the chain rule to get $\int{fδ\int{Vq'dt}dt}$ but that would just allow me to cancel out the $f$ after setting the first variation equal to zero, which doesn't make sense to me, as the problem would be independent from the probability distribution.

Answer 1

Taking the functional to be $$ S[q] = \int_{s_0}^{s_1} f(s) \left( \int_{t_0}^{s} V(q(t), t) \, q'(t) \, dt \right) ds $$ and calculating the functional derivative w.r.t. $q$ I get $$\begin{align} \langle S'[q], \delta q \rangle &= \int_{s_0}^{s_1} f(s) \left( \int_{t_0}^{s} \left( \frac{\partial V(q(t), t)}{\partial q(t)} \delta q(t) \, q'(t) + V(q(t), t) \, \delta q'(t) \right) \, dt\right) ds \\ &= \int_{s_0}^{s_1} f(s) \left( \int_{t_0}^{s} \left( \frac{\partial V(q(t), t)}{\partial q(t)} q'(t) - \frac{dV(q(t), t)}{dt} \right) \delta q(t) \, dt\right) ds \end{align}$$ which suggests the Euler-Lagrange equations be $$ \frac{\partial V(q(t), t)}{\partial q(t)} q'(t) - \frac{dV(q(t), t)}{dt} = 0. $$ But $dV/dt$ can be expanded which results in $$ \frac{\partial V(q(t), t)}{\partial q(t)} q'(t) - \frac{\partial V(q(t), t)}{\partial q(t)} q'(t) - \frac{\partial V(q(t), t)}{\partial t} = 0. $$ Here the first two terms cancel and I'm left with $$ \frac{\partial V(q(t), t)}{\partial t} = 0. $$