Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

Mixed control-state constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measure-valued multipliers in the case of pure state constraints. In particular, numerical solution te...

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Bibliographic Details
Published in:Computational optimization and applications 2010-07, Vol.46 (3), p.511-533
Main Authors: Hoppe, R. H. W., Kieweg, M.
Format: Article
Language:English
Subjects:
Active control
Approximation
Boundary value problems
Constraints
Convex and Discrete Geometry
Discretization
Error analysis
Errors
Estimators
Finite element analysis
Lagrange multiplier
Management Science
Mathematical analysis
Mathematical functions
Mathematical models
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimal control
Optimization
Partial differential equations
Statistics
Studies
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Summary:Mixed control-state constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measure-valued multipliers in the case of pure state constraints. In particular, numerical solution techniques known from the pure control constrained case such as active set strategies and interior-point methods can be used in an appropriately modified way. However, the residual-type a posteriori error estimators developed for the pure control constrained case can not be applied directly. It is the essence of this paper to show that instead one has to resort to that type of estimators known from the pure state constrained case. Up to data oscillations and consistency error terms, they provide efficient and reliable estimates for the discretization errors in the state, a regularized adjoint state, and the control. A documentation of numerical results is given to illustrate the performance of the estimators.
ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-008-9195-4