Finite element method for constrained optimal control problemsgoverned by nonlinear elliptic PDEs

Ming Yan; Lili Chang; Ningning Yan

doi:10.3934/mcrf.2012.2.183

Article Contents

2012, Volume 2, Issue 2: 183-194. Doi: 10.3934/mcrf.2012.2.183

This issue Previous Article Approximate controllability of semilinear reaction diffusion equations Next Article A unified theory of maximum principle for continuous and discrete time optimal control problems

Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs

1.
Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190

Received: February 28, 2011

Revised: October 31, 2011

Published: May 2012

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Abstract

In this paper, we study the finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Instead of the standard error estimates under $L^2$- or $H^1$- norm, we apply the goal-oriented error estimates in order to avoid the difficulties which are generated by the nonsmoothness of the problem. We derive the a priori error estimates of the goal function, and the error bound is $O(h^2)$, which is the same as one for some well known quadratic optimal control problems governed by linear elliptic PDEs. Moreover, two kinds of practical algorithms are introduced to solve the underlying problem. Numerical experiments are provided to confirm our theoretical results.

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Mathematics Subject Classification: Primary: 49J20, 65N15, 65N30.

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