Improved Gradient-Type Algorithms for Optimal Control Problems

Difficulties often arise when we apply the gradient type algorithms employing penalty functions to optimal control problems with variable final time or to problems for which the necessary conditions for optimality require a zero gradient at the final time. In this paper, we first introduce a new transformation method which transforms the variable final time problem into a fixed final time problem. Second, an improved gradient-type algorithm is developed to overcome the zero terminal gradient problem. It is shown that, by applying this transformation and improved algorithm to two examples, not only the variable final time and zero terminal gradient problems are solved and the control vector updated in the correct direction but the rate of convergence of the improved algorithm is faster than that of the traditional gradient-type algorithms.