Optimal control in non-convex domains: a priori discretization error estimates

An optimal control problem for a two-dimensional elliptic equation with pointwise control constraints is investigated. The domain is assumed to be polygonal but non-convex. The corner singularities are treated by a priori mesh grading. Approximations of the optimal solution of the continuous optimal control problem are constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h2.Keywords Linear-quadratic optimal control problems, error estimates, elliptic equations, non-convex domains, corner singularities, control constraints, superconvergence.Mathematics Subject Classification (2000): 49K20, 49M25, 65N30, 65N50

[1]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems , 2005, Comput. Optim. Appl..

[2]  M. Dauge.  Elliptic boundary value problems on corner domains , 1988 .

[3]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem , 2002, Comput. Optim. Appl..

[4]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[5]  J. Roßmann,et al.  Elliptic Boundary Value Problems in Domains with Point Singularities , 2002 .

[6]  Arnd Rösch,et al.  Error estimates for linear-quadratic control problems with control constraints , 2006, Optim. Methods Softw..

[7]  E. N. Dancer.  ELLIPTIC PROBLEMS IN DOMAINS WITH PIECEWISE SMOOTH BOUNDARIES (de Gruyter Expositions in Mathematics 13) , 1996 .

[8]  L. A. Rukhovets,et al.  Variational-difference schemes for linear second-order elliptic equations in a two-dimensional region with piecewise smooth boundary , 1968 .

[9]  T. Geveci,et al.  On the approximation of the solution of an optimal control problem governed by an elliptic equation , 1979 .

[10]  Arnd Rösch,et al.  Superconvergence Properties of Optimal Control Problems , 2004, SIAM J. Control. Optim..

[11]  Fredi Tröltzsch,et al.  Error estimates for linear-quadratic elliptic control problems , 2002, Analysis and Optimization of Differential Systems.

[12]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

[13]  Eduardo Casas,et al.  Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems , 2007, Adv. Comput. Math..

[14]  Arnd Rösch,et al.  Error Estimates for Parabolic Optimal Control Problems with Control Constraints , 2004 .

[15]  Thomas Apel.  Interpolation in h‐Version Finite Element Spaces , 2004 .

[16]  I. Babuška,et al.  Direct and inverse error estimates for finite elements with mesh refinements , 1979 .

[17]  A. H. Schatz,et al.  Maximum norm estimates in the finite element method on plane polygonal domains. II. Refinements , 1979 .

[18]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[19]  Arnd Rösch,et al.  Primal-Dual Active Set Strategy for a General Class of Constrained Optimal Control Problems , 2002, SIAM J. Optim..

[20]  K. Malanowski.  Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems , 1982 .

[21]  Arnd Rösch,et al.  LINFINITY-Estimates for Approximated Optimal Control Problems , 2005, SIAM J. Control. Optim..

[22]  Richard S. Falk,et al.  Approximation of a class of optimal control problems with order of convergence estimates , 1973 .

[23]  Thomas Apel,et al.  Graded Mesh Refinement and Error Estimates for Finite Element Solutions of Elliptic Boundary Value P , 1996 .

[24]  A. H. Schatz,et al.  Maximum norm estimates in the finite element method on plane polygonal domains. I , 1978 .