Error estimate for indirect spectral approximation of optimal control problem governed by fractional diffusion equation with variable diffusivity coefficient

Abstract

In this paper an indirect spectral method of an optimal control problem governed by a space-fractional diffusion equation with variable diffusivity coefficient is studied. First-order optimality conditions of the proposed model are derived and the regularity of the solutions is analyzed. Indirect spectral methods via weighted Jacobi polynomials are built up based on the “first optimize, then discretize” strategy, and a priori error estimates of the discrete optimal control problem in weighted norms are derived. As proposed indirect spectral discrete schemes are designed to accommodate the impact of the variable coefficients, which are complicated and not formulated under the variational framework, conventional error estimate techniques of the discrete space-fractional optimal control problems do not apply. Novel treatments of the discrete variational inequality are developed to resolve the aforementioned issues and to support the error estimates. Numerical examples are presented to verify the theoretical findings.