A finite element method for an elliptic optimal control problem with integral state constraints

Abstract In this article, we discuss a priori error estimates of a bubble enriched nonconforming finite element method for a linear quadratic elliptic distributed optimal control problem with integral state constraints. Therein, using the state equation we reduce the state-control constrained minimization problem to a pure state constrained minimization problem. We have obtained the optimal order (with respect to regularity) error estimates of finite element approximation in two cases: the optimal control problem with integral state constraints together with (i) integral control constraints (ii) pointwise control constraints. Numerical results are presented to confirm the theoretical findings.