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A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. (English) Zbl 1245.65074

A priori error estimates for space-time finite element discretizations of optimal control problems are derived governed by semilinear parabolic parabolic partial differential equations and subject to pointwise control constraints. An approach of D. Meidner and B. Vexler [SIAM J. Control Optim. 47, No. 3, 1150–1177 (2008; Zbl 1161.49026); SIAM J. Control Optim. 47, No. 3, 1301–1329 (2008; Zbl 1161.49035)] is extended where linear-quadratic problems have been considered, discretizing the state equation by usual conforming finite elements in space and a discontinuous Galerkin method in time. Error estimates for controls discretized by piecewise constants functions in time and cellwise constant functions in space are derived in detail, and it is explained how error estimates for further discretization approaches such as cellwise discretization in space, a postprocessing approach of C. Meyer and A. Rösch [SIAM J. Control Optim. 43, No. 3, 970–985 (2004; Zbl 1071.49023)] and the variationally discrete approach of M. Hinze [Comput. Optim. Appl. 30, No. 1, 45–61 (2005; Zbl 1074.65069)] can be obtained. In addition, an estimate for a setting with finitely many time-dependent controls are derived.

MSC:

65K10 Numerical optimization and variational techniques

Software:

GASCOIGNE; RoDoBo
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References:

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