Optimal control of fractional Poisson equation from non-local to local

In this article, the limiting behavior of the solution ${\bar{u}}_s$ of the optimal control problem subjected to the fractional Poisson equation $$(-\mathrm{\Delta }{)}^s{u}_s(x)=f_s(x),x\in \mathrm{\Omega }$$ defined on domain $\mathrm{\Omega }$ bounded by smooth boundary with zero exterior boundary conditions $u_s(x)\equiv 0,x\in {\mathrm{\Omega }}^c$ is established. We will prove that ${\lim }_{s\to 1^{-}}{\bar{u}}_s=\bar{u}$, where $\bar{u}$ is a solution of the optimal control problem subjected to classical Poisson equation $-\mathrm{\Delta }u(x)=f(x),x\in \mathrm{\Omega }$ and $u(x)=0,x\in \mathrm{\partial }\mathrm{\Omega }\mathrm{.}$