An Unconstrained Calculus of Variations Formulation for Generalized Optimal Control Problems and for the Constrained Problem of Bolza

Abstract

This paper has two main ideas. The first idea is that constrained problems in optimal control theory and 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 both the original variables and the multipliers can be explicitly and easily determined. The second idea is that critical point solutions to the associated problem, which include the determination of the multipliers, immediately follows from Euler-Lagrange equations for the unconstrained problem. This critical point solution is a necessary condition for the original problem. We also show that these methods can be combined with previous work by the authors to obtain efficient and accurate numerical solutions to the original problems 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. Finally, we indicate that a large variety of nonclassical optimization problems can be solved by this association.