On the relation between two approaches to exterior penalty method for constrained optimal control problems

The purpose of this paper is to discuss, via the exterior penalty functions method, a class of nonlinear optimal control problems with additional equality and inequality state and control constraints. Two different kinds of penalties are given, in the first the state and control constrained optimal control problem is replaced by a sequence of unconstrained control problems, while the second type transforms the constrained optimal control problem into a sequence of truly unconstrained optimization problems. Two convergence theorems are given to obtain approximate and, in the limit, exact solution of the given constrained optimal control problem. In particular, we show how the necessary conditions of optimality of these two methods yield the familiar Lagrange multipliers of the original constrained optimal control problem in the limit.