An efficient and accurate numerical method for the fractional optimal control problems with fractional Laplacian and state constraint

In this paper, we investigate the numerical approximation of an optimal control problem with fractional Laplacian and state constraint in integral form based on the Caffarelli-Silvestre expansion. The first order optimality conditions of the extended optimal control problem is obtained. An enriched spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is proposed. A priori error estimate for the enriched spectral discrete scheme is proved. Numerical experiments demonstrate the effectiveness of our method and validate the theoretical results.