A spectral Petrov-Galerkin method for optimal control problem governed by a fractional ordinary differential equation

Abstract

In this paper, a spectral Petrov-Galerkin method is investigated for an optimal control problem with a fractional ordinary differential equation constraint. With the study of its well-posedness and the optimality condition in standard Sobolev space, the regularity for the optimal control problem is established in the framework of weighted Sobolev space, which is superior to the counterpart analyzed in standard Sobolev space. Based on the obtained regularity results, the error estimate of the spectral Petrov-Galerkin method is presented, and optimal convergence orders in Lω−α,02 -norm or Lω0,−α2 -norm are determined. Finally, numerical examples with both smooth inputs and rough inputs are given to verify the theoretical prediction.