A fractional-order Taylor wavelet approach for efficient solution of optimal control problems with memory-dependent dynamics

This work investigates the dynamical behavior of optimal control problems involving fractional dynamics using fractional order Taylor wavelets (FOTWs). In the proposed approach, the Caputo fractional derivative together with the Riemann–Liouville fractional integral is employed to efficiently address such complex control problems, allowing for a more accurate description of hereditary and memory features than conventional optimal control models. A key feature of this method is the utilization of FOTWs to approximate both the control and state variables. In this work, the considered model is transformed into a set of algebraic equations using the method of constrained extrema, in which the performance index is adjoined to the constraint equations via Lagrange Multipliers. Furthermore, a convergence analysis is presented, lending credibility to the reliability and effectiveness of the suggested approach. To demonstrate the significance of fractional derivatives, numerical results are reported in both tabular and graphical forms. To establish the accuracy and applicability of the suggested approach, comparisons of approximate errors in the performance index, state variables and control variables with the corresponding available analytical solutions and existing schemes are provided. Moreover, several test problems are investigated to emphasize the accuracy of the scheme and to showcase its potential as a reliable and robust tool for solving fractional optimal control problems.