Some Mathematical Theory of the Penalty Method for Solving Optimum Control Problems

The penalty method is a powerful technique for solving the optimum control problems involving systems subject to holonomic side constraints. In the usual calculus of variations, the above problems are formulated in consideration of the Weierstrass-Erdmann corner conditions which add considerable complexity in practice. In the penalty method, however, the side constraints are eliminated by introducing a sequence of approximate formulations. Thus the Weierstrass-Erdmann corner conditions need not be checked.When the penalty method is applied in the ordinary calculus the sequence of approximate formulations is proved to be equivalent to the original formulation in the limiting case. However, no mathematical rigor has been claimed when the penalty method is applied to the variational problems.The author establishes, in this paper, some mathematical basis for the penalty method applied in the calculus of variations, particularly optimum control problems.