A Haar wavelet-based indirect Galerkin method to solve nonlinear optimal control problems

In this study, we present a novel indirect Galerkin method based on Haar wavelets for solving nonlinear optimal control problems, a domain characterized by its complex and challenging dynamics. By leveraging Pontryagin’s Principle, we derive the first-order necessary conditions and reformulate the problem as a two-point boundary value problem. The proposed method integrates Haar wavelets with the Galerkin method to approximate the state, control, and costate variables, providing a computationally efficient framework. The method benefits from the inherent properties of Haar wavelets, such as their ability to handle sharp discontinuities and abrupt changes, which make them highly suitable for this application.To evaluate the performance of the proposed approach, we validate it against a series of benchmark problems. The numerical results demonstrate superior accuracy and convergence when compared to existing methods, such as the Haar wavelet collocation and pseudospectral approaches. Furthermore, the cost functional values achieved by the method consistently outperform those of previously reported techniques, highlighting its practical efficacy. Despite the computational challenges posed by solving larger systems of equations, the method remains robust and effective across various problem settings. Future research directions include enhancing the theoretical framework by establishing rigorous convergence proofs and error bounds. Overall, this work significantly advances the numerical methodologies available for nonlinear optimal control problems, contributing a powerful tool for addressing real-world applications.