A standard formulation of the one-dimensional variational problem is to find necessary and sufficient conditions for the functional that minimizes

$ \int_0^1 L[t, x_t, \dot x_t] dt $

For a given $L:R^3\rightarrow R$ that is sufficiently well-behaved.

I am encountering applications that result in a slightly different problem:

$ \int_0^1 L[t, x_t, \dot x_t] \left(\int_0^t H[s, x_s, \dot x_s]ds \right) dt $

where $H:R^3\rightarrow R$ can be infinitely differentiable.

I believe this must have either been solved, or it has been covered in some paper; but I couldn't find anything. Perhaps control theorists are very familiar. Any recommendation/pointer/way to attack the problem is welcome.

A standard formulation of the one-dimensional variational problem is to find necessary and sufficient conditions for the functional $x:\mathbb R\rightarrow \mathbb R$ that minimizes

$ \int_0^1 L[t, x_t, \dot x_t] dt $

For a given $L: \mathbb R^3\rightarrow \mathbb R$ that is sufficiently well-behaved.

I am encountering applications that result in a slightly different problem:

$ \int_{[0,1]^2} L[s, t, x_s, x_t, \dot x_s, \dot x_t] dsdt $

I believe this must have either been solved, or it has been covered in some paper; but I couldn't find anything. Perhaps control theorists are very familiar. Any recommendation/pointer/way to attack the problem is welcome.