Smoothing Homotopy Methods for Solving Nonlinear Optimal Control Problems

Abstract

Inspired by the widely used smoothing techniques in image processing and statistical analysis, a novel smoothing homotopy paradigm is proposed in this paper for solving general optimal control problems with the indirect method. In contrast to the existing approaches, the sensitivity associated with the two-point-boundary-value problem (TPBVP) in optimal control is reduced by convoluting appropriate part of the TPBVP with a smoothing kernel. The homotopic process is applied to the bandwidth of the smoothing: a larger bandwidth produces a stronger smoothing effect, rendering the TPBVP better behaved and much easier to solve; a zero bandwidth results in no smoothing and thus leads to the original TPBVP. Depending on whether the Gaussian smoothing kernel or a non-Gaussian kernel is used in the particular implementation, two different smoothing homotopy methods are developed, each with distinctive features of its own. Challenging numerical examples are provided to demonstrate the working and effectiveness of the methods.