Numerical Solution of Two-Dimensional Fractional Optimal Control Problems Using Fractional Vieta-Fibonacci Wavelets

Abstract

This paper presents a novel numerical framework for solving two-dimensional fractional optimal control problems (FOCPs) governed by Caputo fractional partial differential equations. The proposed approach is based on \emph{fractional Vieta--Fibonacci wavelets} (FVFWs), a recently developed wavelet family that combines the recursive structure of Fibonacci polynomials with fractional-order operators. We first construct operational matrices for fractional derivatives in the FVFW basis and employ them to transform the governing FOCP into a system of sparse algebraic equations. The state, adjoint, and control functions are approximated simultaneously in a unified wavelet space, enabling efficient reconstruction of the optimality system. The resulting nonlinear algebraic system is solved using Newton’s method, which guarantees quadratic convergence under standard assumptions. A benchmark two-dimensional fractional control problem is presented to validate the proposed scheme. Numerical results demonstrate that FVFWs achieve high accuracy with relatively few basis functions, outperforming conventional wavelet and spectral approaches in terms of computational efficiency. The method provides a general framework that can be extended to nonlinear fractional PDEs, time-dependent fractional dynamics, and problems with control constraints.