Adaptive Mesh Refinement Method for Optimal Control Using Decay Rates of Legendre Polynomial Coefficients

Abstract

An adaptive mesh refinement method for solving optimal control problems is described. The method employs orthogonal collocation at Legendre-Gauss-Radau points. Accuracy in the method is achieved by adjusting the number of mesh intervals, the polynomial degree within each mesh interval, and, when possible, reducing the mesh size. The decision to increase the degree of the polynomial within a mesh interval or to create new mesh intervals is based on the decay rate of the coefficients of a Legendre polynomial approximation of the state as a function of the index of the Legendre polynomial expansion. The polynomial degree in a mesh interval is increased if the Legendre polynomial coefficient decay rate exceeds a user-specified threshold. Otherwise, the mesh interval is divided into subintervals. The method developed in this brief is then demonstrated on two examples, one of which is a practical problem in aircraft performance optimization. It is found that the approach developed in this brief is more efficient, is simpler to implement, and requires the specification of fewer user-defined parameters when compared with recently developed adaptive mesh refinement methods for optimal control.