On Optimal Control of Quasi-Linear Elliptic Equation with Variable p(x)-Laplacian

We consider an optimal control problem for quasi-linear elliptic equation containing the $p$-Laplacian with variable exponent $p=p(x)$. The exponent $p(x)$ are used as the controls in $L^1(Omega)$. The optimal control problem is to minimize the discrepancy between a given distribution $y_din L^alpha(Omega)$ and the current system state $yin W^{1,p(cdot)}_0(Omega)$ by choosing an appropriate exponent $p(x)$. Basing on the perturbation theory of extremal problems, we study the existence of optimal pairs and propose the ways for relaxation of the original optimization problem.