Optimizing a Functional(?) Where Optimized Quantity is a Function of the Integral Upper Bound?

Question

This may be very simple or asked in an overly obtuse way but I'm struggling to figure out how to find stationary values for $A$ w.r.t. $k$ for $x \in (0,a]$: $$A(x,k)=\int_{0}^{x}f(t,k(t))dt\tag{1}$$ given that $k$ subject to the following constraint: $$\int_{0}^{a} k(t) dt = b.\tag{2}$$ I'm not really sure where to go from here since applying Euler Lagrange seems wrong when $A$ is explicitly a function of the upper bound of the integral rather than just of the functions in the integrand. I assume that if something like the E-L equation can be used then the constraint would be handled with a Lagrange multiplier?

Answer 1

Briefly, the stationary condition becomes

$$ \forall t\in[0,x]:~~\frac{\partial f(k(t),t)}{\partial k(t)}~=~0. $$

For $t\in]x,a]$, pick any values of $k(t)$ such that OP's constraint (2) is satisfied.