# Analysis of control problems of nonmontone semilinear elliptic equations

## TL;DR

This paper investigates optimal control problems for nonmonotone semilinear elliptic equations with convection, establishing existence, uniqueness, regularity, and optimality conditions despite the challenges posed by nonmonotonicity.

## Contribution

It introduces a novel approach using a comparison principle to handle nonmonotone operators, deriving existence, regularity, and optimality conditions for such control problems.

## Key findings

- Existence and uniqueness of solutions proved using comparison principles.
- Regularity and differentiability of control-to-state map established.
- First and second order optimality conditions derived for the control problem.

## Abstract

In this paper we study optimal control problems governed by a semilinear elliptic equation. The equation is nonmonotone due to the presence of a convection term, despite the monotonocity of the nonlinear term. The resulting operator is neither monotone nor coervive. However, by using conveniently a comparison principle we prove existence and uniqueness of solution for the state equation. In addition, we prove some regularity of the solution and differentiability of the relation control-to-state. This allows us to derive first and second order conditions for local optimality.