Approximation of an Optimal Control Problem in the Coefficient for Variational Inequality with Anisotropic p-Laplacian

We study an optimal control problem for a variational inequality with the so-called anisotropic $p$-Laplacian in the principle part of this inequality. The coefficients of the anisotropic $p$-Laplacian, the matrix $A(x)$, we take as a control. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^2(\Omega)$ and the solutions $y\in K\subset W^{1,p}_0(\Omega)$ of the corresponding variational inequality. We show that the original problem is well-posed and derive existence of optimal pairs. Since the anisotropic $p$-Laplacian inherits the degeneracy with respect to unboundedness of the term $|(A(x)\nabla y,\nabla y)_{\mathbb{R}^N}|^{\frac{p-2}{2}}$, we introduce a two-parameter model for the relaxation of the original problem. Further we discuss the asymptotic behavior of relaxed solutions and show that some optimal pairs to the original problem can be attained by the solutions of two-parametric approximated optimal control problems.