Formulation and multigrid solution of Cauchy-Riemann optimal control problems |
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Authors: | M. M. Butt A. Borzì |
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Affiliation: | 1.Institut für Mathematik und Wissenschaftliches Rechnen,Karl-Franzens-Universit?t Graz,Graz,Austria;2.Department of Mathematics,GC University,Lahore,Pakistan;3.Institut für Mathematik,Universit?t Würzburg,Würzburg,Germany |
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Abstract: | 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. |
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