The Galerkin Bell method to solve the fractional optimal control problems with inequality constraints

Abstract

In this manuscript, we adopt the Caputo fractional derivative approach and employ the Galerkin-Bell method to tackle fractional optimal control problems (FOCPs) with equality and inequality constraints in multi-dimensional settings. We derive the Riemann–Liouville (RL) operational matrix for Bell polynomials to facilitate our analysis. By leveraging these matrices and utilizing the Galerkin method, we transform the FOCP into a system of algebraic equations that can be readily solved. We delve into the Bell polynomials’ convergence analysis and error estimation and introduce a residual correction procedure for error estimation. To assess the effectiveness and applicability of our proposed method, we conduct experiments on four different examples and compare our results with those previously reported in the literature. Our findings reveal that the obtained results are highly satisfactory, and in some instances, we achieve the exact solution.