On an Optimal L 1 -Control Problem in Coefficients for Linear Elliptic Variational Inequality

Abstract

We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients $A(x)$ in the main part of the elliptic operator as controls in $L^{\mathrm{1}}(\mathrm{\Omega };{ℝ}^{N(N+\mathrm{1})/\mathrm{2}})$. Since the eigenvalues of such matrices may vanish and be unbounded in $\mathrm{\Omega }$, it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called $H$-admissible solutions.