On infinite-dimensional variational problems

We consider an extremal problem for the functional f(u) = (Latin small letter esh sign)_XΦ(x, u(x), A(u)(x)) μ(dx) where X is a Banach spase, μ is a smooth measure on X, A is a map from a functional space B₁(X) to a functional space B₂(X), and extend the main results of classical calculus of variations to the case under consideration. The infinite-dimensional analogs of the Euler-Lagrange equation, the Noether theorem, the canonical Hamilton system are obtained. The illustrations of these results for A(u)(x) = u′(x) and A(u)(x) = 〈z(x), ∇〉u(x) (where z(x) is a vector field) are given. The example related to the stochastic optimal control theory is considered.