In this paper an optimal control problem is considered, where the control variable lies in a measure space and the state variable fulfills an elliptic equation. This formulation leads to a sparse structure of the optimal control. In this setting we prove a new regularity result for the optimal state and the optimal control. Moreover, a finite element discretization based on [E. Casas, C. Clason, and K. Kunisch, SIAM J. Control Optim., 50 (2012), pp. 1735--1752] is discussed and a priori error estimates are derived, which significantly improve the estimates from that paper. Numerical examples for problems in two and three space dimensions illustrate our results.


  1. optimal control
  2. sparsity
  3. finite elements
  4. error estimates

MSC codes

  1. 35J05
  2. 35Q93
  3. 49M25
  4. 49M29
  5. 65N30
  6. 65N15

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Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 2788 - 2808
ISSN (online): 1095-7138


Submitted: 24 August 2012
Accepted: 13 May 2013
Published online: 2 July 2013


  1. optimal control
  2. sparsity
  3. finite elements
  4. error estimates

MSC codes

  1. 35J05
  2. 35Q93
  3. 49M25
  4. 49M29
  5. 65N30
  6. 65N15



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