A Numerical Method for the Solution of Large Scale State-Constrained Continuous Optimal Control Problems

Abstract

This paper presents a numerical method (penalty function) for the solution of large scale state-constrained continuous optimal control problems. The method is based on the sequential unconstrained minimization of the objective functional augmented by a modified differentiable penalty function, which approximates the solution of the constrained problem. One of the most attractive features of the penalty functions methods is their simplicity i.e. easy implementation from a computational point of view. However, they often lead to an ill-conditioned problem and exhibit slow global convergence. The modified differentiable penalty function presented here overcomes these disadvantages. The overall efficiency of the proposed method also depends on the selection of the unconstrained minimization method. Some of the most efficient methods for unconstrained minimization are: the variable metric (Quasi-Newton) methods and the self-scaling variable metric methods; however there is still some controversy over which of these methods are superior, it has been conjectured that the self-scaling variable metric methods would perform better for highly nonlinear and/or large scale problems. Since the problem studied here is a large scale one, then by comparing the results obtained using a variable metric method for the unconstrained minimization, with those obtained by the corresponding selfscaling versions, the performance of these methods is also assessed attempting to further contribute on the understanding of their behaviour.