Error Estimates for a Class of Elliptic Optimal Control Problems. (2nd January 2017)

• Record Type: Journal Article
• Authors: Mali, O.
• Is Part Of: Numerical functional analysis and optimization. Volume 38:Number 1(2017)
• Page Start: 58
• Page End: 79
• Publication Date: 2017-01-02
• Subjects: Optimal control problems -- elliptic partial differential equations -- error estimates -- duality theory -- numerical methods; 49M29 -- 65K10; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2016.1217881
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.