The Complete Solution for Constrained Delay Problems in the Calculus of Variations by Unconstrained Methods

This paper has two main ideas. The first idea is that general constrained problems with delay in the calculus of variations can be associated with unconstrained calculus of variations problems by using multipliers. This allows us to obtain a true Lagrange multiplier rule where the original variables, the multipliers, and the slack variables for inequality constraints can be determined. The second idea is that critical point solutions to the delay problem, which include the determination of the multipliers, immediately follow from Euler]Lagrange equations for the unconstrained problem. This critical point solution is a necessary condition for the original problem. In later work we will show that these methods can be combined with previous methods by the first author to obtain efficient and accurate numerical solutions to the original problem where no such general numerical methods currently seem to exist. This seems to be an important development since the lack of such methods has hindered the usefulness of the theory.