Abstract and Applied Analysis
Volume 2012 (2012), Article ID 808514, 26 pages
http://dx.doi.org/10.1155/2012/808514
Research Article

Adaptive Finite Element Method for Optimal Control Problem Governed by Linear Quasiparabolic Integrodifferential Equations

Wanfang Shen1,2 and Hua Su2

1School of Mathematics, Shandong University, Jinan 250100, China
2School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China

Received 21 June 2012; Accepted 10 July 2012

Academic Editor: Xinguang Zhang

Copyright © 2012 Wanfang Shen and Hua Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The mathematical formulation for a quadratic optimal control problem governed by a linear quasiparabolic integrodifferential equation is studied. The control constrains are given in an integral sense: 𝑈 𝑎 𝑑 ∫ = { 𝑢 ∈ 𝑋 ; Ω 𝑈 𝑢 ⩾ 0 , 𝑡 ∈ [ 0 , 𝑇 ] } . Then the a posteriori error estimates in 𝐿 ∞ ( 0 , 𝑇 ; 𝐻 1 ( Ω ) ) -norm and 𝐿 2 ( 0 , 𝑇 ; 𝐿 2 ( Ω ) ) -norm for both the state and the control approximation are given.