Approximate solution of nonlinear optimal control problems with scale delay function via a composite pseudospectral approach

In this paper, a direct discretization method is used to approximate the nonlinear scale delayed optimal control problems. The technique is a pseudospectral collocation approach based on the Legendre-Gauss-Lobatto points. Firstly the domain of interest is divided into several adaptive subintervals, and then the traditional pseudospectral approach is used for each segment. This approach discretizes the optimal control problem with a scale delay function and transforms it into a nonlinear programming problem whose solution can be achieved via the existing solvers. The main advantages of this method are the simplicity of the structure and the ease of its implementation and execution. Moreover, the necessary and sufficient conditions of optimality associated with the scale-delayed control problems are obtained. To do this, a new transformation technique is proposed which transforms the scale delayed control problem into a constant delayed one. These conditions can be used to measure the accuracy of the approximate findings obtained by applying the suggested method. The effectiveness and usefulness of the discretization procedure are demonstrated by the implementation of the proposed method in some experimental examples.