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

Abstract

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as the matrix-valued coefficients in L ∞ (Ω; R N ×N ). Given a suitable cost function, the objective is to provide a substantiation of the first order optimality conditions using the concept of convergence in variable spaces. While in the first part [Z. Anal. Anwend. 34 (2015), 85-108] optimality conditions have been derived and analysed in the general case under some assumptions on the quasi-adjoint states, in this second part, we consider diagonal matrices and analyse the corresponding optimality system without such assumptions.

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result

“…In this case (see [12] and [13] ) the first-order optimality conditions can be represented as follows…”