Optimality Conditions for Semilinear Parabolic Equations with Controls in Leading Term

TL;DR

This paper establishes necessary optimality conditions for controlling semilinear parabolic PDEs where the control influences the leading term, using homogenizing spike variation, including cases with state constraints.

Contribution

It introduces a novel approach to derive optimality conditions for PDE control problems with controls in the leading term, extending existing methods to more complex scenarios.

Findings

Derived necessary conditions for optimal controls

Extended results to problems with state constraints

Applied homogenizing spike variation method

Abstract

An optimal control problem for semilinear parabolic partial differential equations is considered. The control variable appears in the leading term of the equation. Necessary conditions for optimal controls are established by the method of homogenizing spike variation. Results for problems with state constraints are also stated.