Derivative with respect a function of a functional with an inner integral

Question

I have to maximize the following functional, which depends on the function $ b(t),\ 0\leq t \leq T$

$Y=\int_0^T [f(\ b(t)\ )\ e^{\int_0^t b(v)^2 dv}]dt$

where b(t) can be supposed to be a "good" function (continuous, etc.). I have thought to derive the functional with respect to $b(\cdot)$ (then null the equation and solve it), but I don't know how to deal with the inner integral. Any idea on how to do it? Do you have any good references on maximization of functionals?

Answer 1

The substitution

$u(t) = \int_0^t b(v)^2 dv $

gives $b(t) = \sqrt{u'(t)}$ (let's suppose $b(t) \geq 0$). The functional becomes

$Y = \int_0^T f \left( \sqrt{u'(t)} \right ) e^{u(t)} dt$

and it is possible to use variational calculus.