Hybrid RBF Method for Solving Fractional PDE-Constrained Optimal Control Problems

This study addresses the numerical solution of an optimal control problem governed by a fractional convection–reaction–diffusion partial differential equation. The approach utilizes Radial Basis Function–Partition of Unity (RBF-PU) methods combined with the Grünwald-Letnikov approximation for fractional derivatives, which provides a fundamental extension of classical derivatives in fractional calculus.‎ To enhance sparsity in the control strategy, an L_۲ norm is integrated into the objective function, along with quadratic penalties to reduce deviations from the desired state. This hybrid formulation facilitates the effective management of spatially sparse controllers, relevant in many practical applications. ‎ The RBF-PU technique offers a flexible and efficient framework by partitioning the domain into overlapping subregions, applying local RBF approximations, and synthesizing the global solution with compactly supported weight functions. Numerical experiments demonstrate the accuracy and effectiveness of this method.

کلیدواژه ها:

Local RBF approximation‎ ، ‎Heat equation‎ ، ‎Fractional derivative‎ ، Optimal control‎ ، ‎L_۲ norm