Optimal Control in Matrix-Valued Coefficients for Nonlinear Monotone Problems: Optimality Conditions I

Abstract

In this article we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as matrix-valued coefficients in $L^{\infty }(\Omega ;{\mathbb{R}}^{N\times N})$. For the exemplary case of a tracking cost functional, we derive first order optimality conditions. This first part is concerned with the general case of matrix-valued coefficients under some hypothesis, while the second part focuses on the special class of diagonal matrices.