Penalty functions and singular arcs in optimal steel refining

In a recent paper by Woodside et al. [1] the introduction of a penalty function to handle state constraints was found to lead to a singular arc in a problem of optimizing an electric steel refining process. Some numerical results were also given indicating a slow convergence of the penalty function solution to the solution of a constrained problem. In this correspondence a slightly different penalty function is used for the same problem. It is shown that the penalty function solution not only roughly approximates the solution to the state constrained problem, but that the singular arc and state constrained arcs coincide in the limit.