Error estimates for the FEM approximation of optimal sparse control of elliptic equations with pointwise state constraints and finite‐dimensional control space

Abstract

Summary In this work, we derive an a priori error estimate of order h2|log(h)| for the finite element approximation of a sparse optimal control problem governed by an elliptic equation, which is controlled in a finite dimensional space. Furthermore, box‐constrains on the control are considered and finitely many pointwise state‐constrains are imposed on specific points in the domain. With this choice for the control space, the achieved order of approximation for the optimal control is optimal, in the sense that the order of the error for the optimal control is of the same order of the approximation for the state equation.