A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part I: Problems Without Control Constraints

Abstract

In this paper we develop a priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters. The paper is divided into two parts. In the first part we develop some stability and error estimates for space-time discretization of the state equation and provide error estimates for optimal control problems without control constraints. In the second part of the paper, the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable.

MSC codes

  1. 49N10
  2. 49M25
  3. 65M15
  4. 65M60

Keywords

  1. optimal control
  2. parabolic equations
  3. error estimates
  4. finite elements

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Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 1150 - 1177
ISSN (online): 1095-7138

History

Submitted: 8 June 2007
Accepted: 27 November 2007
Published online: 19 March 2008

MSC codes

  1. 49N10
  2. 49M25
  3. 65M15
  4. 65M60

Keywords

  1. optimal control
  2. parabolic equations
  3. error estimates
  4. finite elements

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