On spectral Petrov-Galerkin method for solving optimal control problem governed by a two-sided fractional diffusion equation

TL;DR

This paper develops a spectral Petrov-Galerkin method for solving an optimal control problem governed by a two-sided fractional diffusion equation, addressing boundary singularities and providing error estimates and an efficient solution algorithm.

Contribution

It introduces a novel spectral Petrov-Galerkin approach for fractional PDE-constrained optimal control problems, including regularity analysis, error estimates, and a fast gradient algorithm.

Findings

Error estimates confirm convergence orders for state and control variables.

Numerical experiments validate theoretical error bounds.

The proposed algorithm achieves quasi-linear complexity and efficiency.

Abstract

In this paper, we investigate a spectral Petrov-Galerkin method for an optimal control problem governed by a two-sided space-fractional diffusion-advection-reaction equation. Taking into account the effect of singularities near the boundary generated by the weak singular kernel of the fractional operator, we establish the regularity of the problem in weighted Sobolev space. Error estimates are provided for the presented spectral Petrov-Galerkin method and the convergence orders of the state and control variables are determined. Furthermore, a fast projected gradient algorithm with a quasi-linear complexity is presented to solve the resulting discrete system. Numerical experiments show the validity of theoretical findings and efficiency of the proposed fast algorithm.