A fully direct transcription method for solving distributed-order time-fractional diffusion optimal control problems with unilateral constraints

Abstract

This paper explores optimal control problems governed by distributed-order time-fractional diffusion equation, with a particular emphasis on those involving unilateral constraints. The primary goal is to establish the existence and uniqueness of the solution, derive the necessary optimality conditions, and subsequently obtain approximate solutions using direct methods. In the numerical direct method, the spatial derivative is approximated using finite difference formulas, while the distributed-order time-fractional derivative is approximated using derivative operational matrices based on the Grünwald-Letnikov and L1 methods. Furthermore, the performance index is approximated through a suitable quadrature rule for improved accuracy. As a result, the problem is transformed into a convex quadratic optimization problem, which can be solved efficiently using well-established quadratic optimization algorithms. To assess the precision and effectiveness of our approach, we conducted numerical experiments on three distinct examples, encompassing cases both with and without unilateral constraints. In scenarios with exact solutions, our method consistently produced results closely aligning with them. For cases lacking exact solutions, the approximations highlighted the robustness of the method. These evaluations emphasize the method’s applicability and reliability.