Discretization estimates for an elliptic control problem

• Published in: Numerical functional analysis and optimization Vol. 19; no. 5-6; pp. 431 - 464
• Main Authors: ARNAUTU, V, NEITTAANMÄKI, P
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Marcel Dekker, Inc 01.01.1998; Taylor & Francis
• Subjects: Calculus of variations and optimal control; elliptic equation; error estimates; Exact sciences and technology; finite element method; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; optimal control; Partial differential equations; Partial differential equations, boundary value problems; Sciences and techniques of general use; spectral method; two-point BVP; Galerkin-Petrov method; Ritz Galerkin method; Elliptic equation; Error estimation; Differential equation; Variational calculus; Optimality condition; Optimal control (mathematics); Partial differential equation; Discretization method
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1080/01630569808816838

An optimal control problem governed by an elliptic equation written in variational form in an abstract functional framework is considered. The control is subject to restrictions. The optimality conditions are established and the Ritz-Galerkin discretization is introduced. If the error estimate corresponding to the elliptic equation is given as a function like where h is the discretization parameter and is an integer, then the error estimates for the optimal control, for the optimal state and for the optimal value are obtained. These results are applied first for a Two-Point BVP and next for a 2D/3D elliptic problem as state equation. Next a spectral method is used in the discretization process. The estimates obtained in the abstract case are applied to a distributed control problem and to a boundary control problem.