Cesari-type Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls

TL;DR

This paper investigates optimal control problems for semilinear elliptic PDEs with control-dependent leading terms, establishing existence results via Cesari-type conditions and analyzing the $G$-closure of the leading coefficient.

Contribution

It introduces a novel approach using Cesari-type conditions to prove existence of solutions for control problems with control-dependent leading terms in elliptic equations.

Findings

Existence of solutions under Cesari-type conditions.

Analysis of the $G$-closure of the leading term.

Framework applicable to control problems with divergence form equations.

Abstract

An optimal control problem governed by semilinear elliptic partial differential equations is considered. The equation is in divergence form with the leading term containing controls. By studying the $G$-closure of the leading term, an existence result is established under a Cesari-type condition.