A pseudo−operational collocation method for optimal control problems of fractal−fractional nonlinear Ginzburg−Landau equation

Document Type : Research Article

Authors

• T. Shojaeizadeh ¹
• E. Golpar-Raboky ²
• Parisa Rahimkhani ³

¹ Department of Mathematics, Qom Branch, Islamic Azad University, Qom, Iran.

² Department of Mathematics, University of Qom, Qom, Iran.

³ Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran.

10.22067/ijnao.2024.86362.1375

Abstract

The presented work introduces a new class of nonlinear optimal control problems in two dimensions whose constraints are nonlinear Ginzburg−Landau equations with fractal−fractional (FF) derivatives. To acquire their ap-proximate solutions, a computational strategy is expressed using the FF derivative in the Atangana−Riemann−Liouville (A-R-L) concept with the Mittage-Leffler kernel. The mentioned scheme utilizes the shifted Jacobi polynomials (SJPs) and their operational matrices of fractional and FF derivatives. A method based on the derivative operational matrices of SJR and collocation scheme is suggested and employed to reduce the problem into solving a system of algebraic equations. We approximate state and control functions of the variables derived from SJPs with unknown coef-ficients into the objective function, the dynamic system, and the initial and Dirichlet boundary conditions. The effectiveness and efficiency of the suggested approach are investigated through the different types of test problems.

Keywords

• Fractal−fractional (FF) derivative
• Shifted Jacobi polynomials (SJPs)
• Operational matrices
• Nonlinear Ginzburg−Landau equation
• Opti- mal control problem

Main Subjects

• Control and Optimization

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