On direct methods for solving variational problems

In the paper, new “direct” methods for solving problems of Calculus of Variations are described. The methods suggested are based on reduction (by means of exact penalty functions) of the initial constrained minimization problem to an unconstrained minimization one. The constraints may be of different nature (boundary conditions, differential or integral relations etc.). A comparison with Ritz' and Galerkin's methods is given. The proposed approach can be useful for practical solution of constrained problems in Control Theory and Calculus of Variations.