International Journal of Nonlinear Analysis and Applications

A hybrid polynomial operational matrix method for optimal control problems

Articles in Press

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Qom, Qom, Iran

2 Mathematics Department, College of Education for Pure Science, University of Babylon, Hilla 51001, Iraq

Abstract

This paper presents a novel computational method for solving optimal control problems using a hybrid operational matrix framework. The approach is based on a convex combination of Legendre, Chebyshev, and Bernstein polynomials, which synergistically leverages the superior approximation properties of Legendre bases, the boundary-clustering advantage of Chebyshev polynomials, and the numerical stability of Bernstein bases. The Pontryagin necessary optimality conditions are transformed into a system of algebraic equations via the constructed hybrid operational matrix of differentiation, enabling an efficient and accurate numerical solution. The proposed method is validated on several benchmark problems with known analytical solutions. Numerical results demonstrate that the hybrid framework achieves machine-precision accuracy and exhibits exponential convergence, outperforming many existing single-basis methods. This work establishes that combining multiple polynomial families creates a more robust, flexible, and highly accurate numerical tool for optimal control.

Keywords

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International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 28 April 2026
  • Receive Date: 11 December 2025
  • Accept Date: 16 February 2026