An efficient optimization algorithm for nonlinear 2D fractional optimal control problems

• Published in: The Journal of supercomputing Vol. 80; no. 6; pp. 7906 - 7930
• Main Authors: Moradikashkooli, A., Haj Seyyed Javadi, H., Jabbehdari, S.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-04-2024; Springer Nature B.V
• Subjects: Algorithms; Basis functions; Compilers; Computer Science; Dynamical systems; Interpreters; Lagrange multiplier; Mathematical analysis; Optimal control; Optimization; Optimization algorithms; Polynomials; Processor Architectures; Programming Languages; Quadratures; System dynamics; Wave equations; Operational matrix; Nonlinear 2D fractional optimal control problems; Optimization algorithm; Coefficients and parameters; Generalized Laguerre polynomials
• ISSN: 0920-8542; 1573-0484
• DOI: 10.1007/s11227-023-05732-z

In this research article, we present an optimization algorithm aimed at finding the optimal solution for nonlinear 2-dimensional fractional optimal control problems that arise from nonlinear fractional dynamical systems governed by Caputo derivatives under Goursat–Darboux conditions. The system dynamics are described by equations such as the Klein–Gordon, convection–diffusion, and diffusion–wave equations. Our algorithm utilizes a novel class of basis functions called generalized Laguerre polynomials (GLPs), which are an extension of the traditional Laguerre polynomials. To begin, we introduce the GLPs and their properties, and we develop several new operational matrices specifically tailored for these basis functions. Next, we expand the state and control functions using the GLPs, with the coefficients and control parameters remaining unknown. This expansion allows us to transform the original problem into an algebraic system of equations. To facilitate this transformation, we employ operational matrices of Caputo derivatives, the rule of 2D Gauss–Legendre quadrature, and the method of Lagrange multipliers.