Matrix-Valued $L^1$-Optimal Controls in the Coefficients of Linear Elliptic Problems

Abstract

We consider optimal control problems for linear degenerate elliptic equations with mixed boundary conditions. In particular, we take the \vspace{-.6mm} matrix-valued coefficients $A(x)$ of such systems as controls in $L^1(\Omega ;{\mathbb{R}}^{\frac{N(N+1)}{2}})$. One of the important features of the admissible controls is the fact that eigenvalues of the coefficient matrices may vanish in $\Omega$. Equations of this type may exhibit non-uniqueness of weak solutions. Using the concept of convergence in variable spaces and following the direct method in the Calculus of variations, we establish the solvability of this optimal control problem in the class of weak admissible solutions.