A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems

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 different...

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Bibliographic Details
Published inJournal of optimization theory and applications Vol. 178; no. 2; pp. 317 - 348
Main Authors Ludovici, Francesco, Neitzel, Ira, Wollner, Winnifried
Format Journal Article
LanguageEnglish
Published New York Springer US 01.08.2018
Springer Nature B.V
Subjects
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Discretization
Engineering
Estimates
Finite element method
Mathematical analysis
Mathematics
Mathematics and Statistics
Numerical analysis
Operations Research/Decision Theory
Optimal control
Optimization
Partial differential equations
State variable
Theory of Computation
65M15
Optimal control
State constraints
Pointwise in time constraints
65M60
Space-time a priori error estimates
49M25
65M12
Semilinear parabolic PDE
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Summary: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.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-018-1311-8