Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls

Hongwei Lou; Jiongmin Yong

doi:10.3934/mcrf.2018003

Article Contents

2018, Volume 8, Issue 1: 57-88. Doi: 10.3934/mcrf.2018003

This issue Previous Article Optimal voltage control of non-stationary eddy current problems Next Article Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes

Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls

Hongwei Lou^1, and
Jiongmin Yong^2,

1.
School of Mathematical Sciences, and LMNS, Fudan University, Shanghai 200433, China,
2.
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

Received: March 2017

Revised: October 2017

Published: January 2018

The first author was supported in part by NSFC grant 11371104, the second author was supported in part by NSF grant DMS-1406776.

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Abstract

An optimal control problem for a semilinear elliptic equation of divergenceform is considered. Both the leading term and the semilinear term of the state equationcontain 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.

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Mathematics Subject Classification: Primary: 49K20, 35J61; Secondary: 35Q93.

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