Second-Order Necessary Conditions for Optimal Control of Semilinear Elliptic Equations with Leading Term Containing Controls

TL;DR

This paper investigates second-order necessary conditions for optimal control problems involving semilinear elliptic equations where controls appear in both the leading and nonlinear terms, extending the classical maximum principle.

Contribution

It develops second-order necessary conditions for control problems with controls in the leading term of semilinear elliptic equations, addressing singular cases where first-order conditions are insufficient.

Findings

Derived second-order necessary conditions for singular controls.

Extended the Pontryagin maximum principle to second-order in this context.

Provided theoretical framework for analyzing optimal controls with controls in the leading term.

Abstract

An optimal control problem for a semilinear elliptic equation of divergence form is considered. Both the leading term and the semilinear term of the state equation contain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.