Weak Optimal Controls in Coefficients for Linear Elliptic Problems

TL;DR

This paper investigates an optimal control problem for linear degenerate elliptic equations with mixed boundary conditions, focusing on the solvability and non-uniqueness issues when controlling via weight functions in $L^1( abla)$.

Contribution

It introduces a novel control framework using weight functions in $L^1( abla)$ for degenerate elliptic equations and analyzes solvability with weak solutions.

Findings

Addresses Lavrentieff phenomenon and non-uniqueness in solutions.

Establishes solvability conditions for the control problem.

Utilizes the direct method in the Calculus of variations.

Abstract

In this paper we study an optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. The equations of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions. We adopt the weight function as a control in $L^1(\Omega)$. Using the direct method in the Calculus of variations, we discuss the solvability of this optimal control problem in the class of weak admissible solutions.