A hp-h Adaptive Time-mesh Refinement Algorithm for Solving Optimal Control Problems

Abstract

We propose a new iterative hp-h Adaptive Mesh Refinement (hp-h AMR) algorithm for solving continuous-time optimal control problems. We start by applying a hp-method based on the orthogonal collocation with Legendre- Gauss-Radau points using both a pre-defined fixed number of sub-intervals and also a fixed polynomial degree at each interval. By varying the number of mesh intervals and the polynomial degree in each mesh interval, the accuracy of method can be tuned. For that, we consider a h-tolerance and a p-tolerance. The decision to increase or decrease the degree of the polynomial in each segment is based on the p-tolerance. Moreover, the mesh interval is divided into subintervals considering different levels of refinement based on the h-tolerance. This iterative hp-h AMR procedure stops when both h-tolerance and p-tolerance criteria are satisfied. We illustrate this algorithm by solving an optimal control problem involving a nonholonomic car-like system with state constraints which is characterized by presenting strong nonlinearities and by having discontinuous controls.