Let's say we have the following function ;
$\intop_{0}^{\infty}\int_{0}^{N}V\left(C(t,\tau\right)dtd\tau$
and we want to maximise it according to the following constraint ;
$\dot{K}\left(t\right)=F\left(K\left(t\right)\right)-C\left(t,\tau\right)$
Note that $t$ is time and $\tau$ is age. $V$ is a function in terms of $C(t,\tau)$
How could it be possible to maximize this function under this constraint ? I looked a little bit on some documents on calculus of variation with two-dimensional problems but I could not find a result.