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A numerical method for solving a special class of optimal control problems is given. The solution is based on state parametrization as a polynomial with unknown coefficients. This converts the problem to a non-linear optimization problem. To facilitate the computation of optimal coefficients, an improved iterative method is suggested. Convergence of this iterative method and its implementation for numerical examples are also given.

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Recently, a mesh refinement strategy is presented on pseudospectral methods for solving optimal control problems by using the relative curvature of the state approximation to choose the type of discretization change in each iteration. Nevertheless, this criterion requires a large amount of computational cost in terms of CPU time. The main goal of this paper is to draw attention to select a suitable criterion with fewer computational cost. To this end, we use the arc length of the state approximation in the mesh interval based on the relative error estimate that was recently provided. We also update the number of mesh intervals and the location of mesh points according to the behaviour of the arc length. Indeed, by implementing this criterion, we do not need to solve an optimization problem anymore, and so significantly reduce the computational time as well as CPU times. Finally, we illustrate the accuracy, efficiency and ability of the arc length criterion in comparison with the curvature by offering some numerical examples.

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