Generalized Gaussian smoothing homotopy method for solving nonlinear optimal control problems

Abstract

This paper introduces an innovative generalized Gaussian smoothing homotopy method for solving nonlinear optimal control problems using the indirect method. Compared to the original smoothing homotopy methods, this approach leverages a multivariate Gaussian function to smooth both state and costate variables, extending the convolution process beyond the time domain to all unknown variables. By utilizing the separability property of the Gaussian kernel, the multivariate convolution is decomposed into univariate convolutions along each dimension, allowing independent and efficient computation. Additionally, the Gauss–Chebyshev quadrature technique is employed to approximate these univariate convolutions, further reducing computational complexity. The convergence of the method is demonstrated through challenging numerical examples, showcasing its superiority over Gaussian smoothing homotopy method.