Direct approximate methods

of solving variational problems for non-local functionals

(survey)

G.A.Kamenskiy

The non-local functional is an integral with the integrand depending on the unknown function at different values of arguments. These types of functionals have different applications in physics, engineering and sciences. There are described here the variational problems for non-local functionals and the direct methods for approximate solution of these problems.

The first studies of variational problems for non-local functionals were made by L.E.Elsgolts. He considered the asymmetrical problem. Later the asymmetrical problem was investigated by author, and the symmetrical variational problem for non-local functionals was studied. Also variational problems for non-local functionals with many deviations of the argument are studied љand for non-local functionals depending on functions of many arguments with deviations this problem is investigated. Variational problems for functional depending on functions of two arguments were studied also by author.

The theory of mixed functional differential equations was intensively developed in the last years. Here variational problems for mixed non-local functionals was studied by author with colleagues. Another approach to the variational problems for non-local functional is published by

M.Drakhlin, E.Litsyn, E.Stepanov. Also an important problem of damping a control system was solved by the variational method in these works.

Analytical methods of solving variational and boundary value problems for functional differential equations are effective only in the simple cases. Therefore a great role play approximate methods of solution of variational problems for non-local functionals and corresponding boundary value problems. There are some works describing the approximate methods of solution of boundary value problems connected with the variational problems for non-local functionals, but this paper is dedicated to review of the direct approximate methods of solving the variational problems for non-local functionals. There are described the method of local variations and the direct method of Euler in application to different types of variational problems for non-local functionals.