A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems
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We consider the finite element discretization of semilinear parabolic optimization problems subject to pointwise in time constraints on mean values of the state variable. In order to control the feasibility violation induced by the discretization, error estimates for the semilinear partial differential equation are derived. Based upon these estimates, it can be shown that any local minimizer of the semilinear parabolic optimization problems satisfying a weak second-order sufficient condition can be approximated by the discretized problem. Rates for this convergence in terms of temporal and spatial discretization mesh sizes are provided. In contrast to other results in numerical analysis of optimization problems subject to semilinear parabolic equations, the analysis can work with a weak second-order condition, requiring growth of the Lagrangian in critical directions only. The analysis can then be conducted relying solely on the resulting quadratic growth condition of the continuous problem, without the need for similar assumptions on the discrete or time semidiscrete setting.
KeywordsOptimal control Semilinear parabolic PDE State constraints Pointwise in time constraints Space-time a priori error estimates
Mathematics Subject Classification49M25 65M12 65M15 65M60
The authors are grateful for the support of their former host institutions. To this end, I. Neitzel acknowledges the support of the Technische Universität München and F. Ludovici and W. Wollner the support of the Universität Hamburg.
- 21.Bonnans, J., Shapiro, A.: Perturbation Analysis of Optimization Problems, Springer Series Operations Research Financial Engineering, 1st edn. Springer, New York (2010)Google Scholar
- 28.Ludovici, F.: Numerical analysis of parabolic optimal control problems with restrictions on the state and its first derivative. Ph.D. thesis, Technische Universität Darmstadt (2017). http://tuprints.ulb.tu-darmstadt.de/6781/
- 31.Thomèe, V.: Galerkin Finite Element Methods for Parabolic Problems, Springer Ser. Comput. Math., 2nd edn. Springer, Berlin (2006)Google Scholar