Suppose I want to choose a function $a$ to maximise the following infinite dimensional linear program:
$$ \int_0^1 x a(x) dx $$
subject to the constraints
$$ \int_0^x a(t)dt = k(x)$$
for some given function $k(x)$, for all $x \in [0,1]$.
Is it possible to use Lagrangian techniques to solve this? Are there complications involved because of the continuum of equality constraints?
Is $k(x)$ differentiable? If yes, $a(x)=k(x)'+c$, where $c$ is an arbitrary constant. Now $\int_0^{1} xa(x)dx = \int_{0}^{1}xk'(x)dx + c*\frac{1}{2} = k(1) - \int_{0}^{1}k(x)dx+c*\frac{1}{2}$, which all depends on how we choose the constant $c$.