A spectral collocation method via Krawtchouk polynomials for two-dimensional Liouville–Caputo fractional optimal control problems

This paper introduces an efficient spectral collocation method for solving two-dimensional fractional optimal control problems involving Liouville–Caputo derivatives. The proposed approach employs a Krawtchouk polynomials basis, taking advantage of its discrete orthogonality and strong localization properties. An operational matrix for the Riemann–Liouville fractional integral is developed, enabling accurate and efficient handling of the non-local memory effect. By approximating the state and control variables with Krawtchouk polynomials, the original problem is transformed into an algebraic system solved using Newton’s method. Theoretical convergence analysis and numerical stability tests confirm the method’s reliability. Numerical experiments demonstrate its high accuracy, computational efficiency, and robustness. These results indicate that the Krawtchouk polynomials framework is a promising tool for modeling and controlling complex fractional-order systems in science and engineering.