On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions

Shu Luan

doi:10.3934/mcrf.2017018

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2017, Volume 7, Issue 3: 493-506. Doi: 10.3934/mcrf.2017018

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On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions

Shu Luan^,

School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China

* Corresponding author:Shu Luan
* Corresponding author:Shu Luan

Received: May 2016

Revised: January 2017

Published: October 2017

This work was partially supported by the National Natural Science Foundation of China under grants 11301472 and 11601213, the Natural Science Foundation of Guangdong Province under grant 2014A030307011, the Training Program Project for Outstanding Young Teachers of Colleges and Universities in Guangdong Province under grant Yq2014116 and the Characteristic Innovation Project of Common Colleges and Universities in Guangdong Province under grant 2014KTSCX158.

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Abstract

An optimal control problem governed by a class of semilinear elliptic equations with nonlinear Neumann boundary conditions is studied in this paper. It is pointed out that the cost functional considered may not be convex. Using a relaxation method, some existence results of an optimal control are obtained.

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Mathematics Subject Classification: Primary: 49J20; Secondary: 35J66.

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