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A posteriori error estimates of spectral method for the fractional optimal control problems with non-homogeneous initial conditions

In this paper we consider an optimal control problem governed by a space-time fractional diffusion equation with non-homogeneous initial conditions. A spectral method is proposed to discretize the problem in both time and space directions. The contribution of the paper is threefold: (1) A discussion and better understanding of the initial conditions for fractional differential equations with Riemann-Liouville and Caputo derivatives are presented. (2) A posteriori error estimates are obtained for both the state and the control approximations. (3) Numerical experiments are performed to verify that the obtained a posteriori error estimates are reliable.