Mesh Refinement Using Density Function for Solving Optimal Control Problems

This paper presents a general mesh refinement method based on a density function for distributing the grid points, which can be used for solving optimal control problems through a direct method. The algorithm captures any discontinuities and smoothness irregularities in the state and control variables with high resolution, thus helping with the convergence of the overall optimization algorithm. The algorithm also provides flexibility during the mesh refinement process, especially for problems with multiple control inputs. As a specific example of the general theory, a density function based on the best piecewise linear approximation of functions with curvature defined almost everywhere is proposed for refining the mesh. The technique is applied to solve the problem of the optimal powerlimited landing for a large commercial aircraft.