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In this paper, we propose a Galerkin finite element method for the ellipticoptimal control problem governed by the Riesz space-fractional PDEs on 2Ddomains with control variable being discretized by variational discretizationtechnique. The optimality condition is derived and priori error estimates ofcontrol, costate and state variables are successfully established. Numericaltest is carried out to illustrate the accuracy performance of this approach.

A Galerkin FE method for elliptic optimal control problem governed by 2D space-fractional PDEs