Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems

• Published in: Numerical algorithms Vol. 65; no. 1; pp. 171 - 194
• Main Authors: Saberi Nik, H., Effati, S., Motsa, S. S., Shirazian, M.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 2014; Springer Nature B.V
• Subjects: Algebra; Algorithms; Boundary value problems; Chebyshev approximation; Collocation; Computer Science; Convergence; Homotopy theory; Iterative algorithms; Iterative methods; Mathematical models; Matlab; Maximum principle; Nonlinear control; Nonlinearity; Numeric Computing; Numerical Analysis; Optimal control; Original Paper; Perturbation methods; Spectra; Theory of Computation; Pontryagin’s maximum principle; Spectral homotopy analysis method; Spectral collocation; Optimal control problems
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-013-9700-4

A combination of the hybrid spectral collocation technique and the homotopy analysis method is used to construct an iteration algorithm for solving a class of nonlinear optimal control problems (NOCPs). In fact, the nonlinear two-point boundary value problem (TPBVP), derived from the Pontryagin’s Maximum Principle (PMP), is solved by spectral homotopy analysis method (SHAM). For the first time, we present here a convergence proof for SHAM. We treat in detail Legendre collocation and Chebyshev collocation. It is indicated that Legendre collocation gives the same numerical results with Chebyshev collocation. Comparisons are made between SHAM, Matlab bvp4c generated results and results from literature such as homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM) and differential transformations.