Computing and Visualization in Science

, Volume 14, Issue 2, pp 79–90 | Cite as

Formulation and multigrid solution of Cauchy-Riemann optimal control problems

  • M. M. Butt
  • A. BorzìEmail author


The formulation of optimal control problems governed by Cauchy-Riemann equations is presented. A distributed control mechanism through divergence and curl sources is considered with the boundary conditions of mixed type. A Lagrange multiplier framework is introduced to characterize the solution to Cauchy-Riemann optimal control problems as the solution of an optimality system of four first-order partial differential equations and two optimality conditions. To solve the optimality system, staggered grids and multigrid methods are investigated. It results that staggered grids provide a natural collocation of the optimization variables and second-order accurate solutions are obtained. The proposed multigrid scheme is based on a coarsening by a factor of three that results in a nested hierarchy of staggered grids. On these grids a distributed-Gauss-Seidel and gradient-based smoothing scheme is employed. Results of numerical experiments validate the proposed optimal control formulation and demonstrate the effectiveness of the staggered-grids multigrid solution procedure.


Cauchy-Riemann equations Optimal control problems Staggered grids Multigrid methods 

Mathematics Subject Classification (2000)

35Q93 49K20 65N55 


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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institut für Mathematik und Wissenschaftliches RechnenKarl-Franzens-Universität GrazGrazAustria
  2. 2.Department of MathematicsGC UniversityLahorePakistan
  3. 3.Institut für MathematikUniversität WürzburgWürzburgGermany

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