Numerical analysis of a family of optimal distributed control problems governed by an elliptic variational inequality

The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter $\alpha >0$) is obtained through the finite element method when its parameter $h\to 0$. We also obtain the limit of the discrete optimal control and the associated state system solutions when $\alpha \to \mathrm{\infty }$ (for each $h>0$) and a commutative diagram for two continuous and two discrete optimal control and its associated state system solutions is obtained when $h\to 0$ and $\alpha \to \mathrm{\infty }$. Moreover, the double convergence is also obtained when $(h,\alpha )\to (0,\mathrm{\infty })$.