I am trying to solve a problem that I think I should use calculus of variations to solve (maybe optimal control?). I am trying to figure out the correct technique to use, or to figure out whether this is solvable at all. If it is solvable, I am happy to do the work of learning what I need to know in order to solve it.
Simple (solvable) problem:
The simple problem that I can solve is the following:
$\min_{p(\theta)}\int_{\theta=0}^{1}(g[p(\theta)]+T(\theta))f(\theta)d\theta$
where $T(\theta)=\int_\theta^1(p(x)-1)dx-\theta p(\theta)+\theta$. Substituting in yields:
$\min_{p(\theta)}\int_{\theta=0}^{1}(g[p(\theta)]+\int_\theta^1(p(x)-1)dx-\theta p(\theta)+\theta)f(\theta)d\theta$
Integrating by parts I get:
$\min_{p(\theta)}\int_{0}^{1}\left[\left[g[p(\theta)]-\theta p(\theta)+\theta \right]f(\theta) + F(\theta)[1-p(\theta)] \right]d\theta$
And I can then minimize pointwise with respect to $p(\theta)$ for all $\theta$. Yielding the function $p(\theta)$ that minimizes the argument.
Hard problem:
Now I would like to generalize this to a slightly different function:
$\min_{p(\theta)}\int_{\theta=0}^{1}(g[p(\theta)]+h[T(\theta)])f(\theta)d\theta$.
I now have a function, $h[T(\theta)]$ rather than just $T(\theta)$. Substituting in $T(\theta)$, I get stuck when I get to:
$\min_{p(\theta)}\int_{\theta=0}^{1}(g[p(\theta)]+h[1 -\theta p(\theta)+ \int_\theta^1p(x)dx])f(\theta)d\theta$.
The fact that $\int_\theta^1p(x)dx$ is inside of the function $h()$ prevents me from integrating by parts, and getting rid of it.
I am trying to figure out whether this type of function is solvable. I don't know much about calculus of variations (or other approaches that might be used), but I would be willing to learn whatever techniques could help me solve this problem.
Any leads that you could give me would be greatly appreciated! Thanks!
$p(\theta):\theta\in[0,1]\rightarrow [0,1]$. $g()$ and $h()$ are increasing and convex.