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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| Published in | Journal of optimization theory and applications Vol. 178; no. 2; pp. 317 - 348 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.08.2018
Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 0022-3239 1573-2878 |
| DOI | 10.1007/s10957-018-1311-8 |
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| Abstract | 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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| AbstractList | 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. |
| Author | Ludovici, Francesco Wollner, Winnifried Neitzel, Ira |
| Author_xml | – sequence: 1 givenname: Francesco surname: Ludovici fullname: Ludovici, Francesco organization: Fachbereich Mathematik, Technische Universität Darmstadt – sequence: 2 givenname: Ira surname: Neitzel fullname: Neitzel, Ira organization: Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn – sequence: 3 givenname: Winnifried orcidid: 0000-0002-6571-8043 surname: Wollner fullname: Wollner, Winnifried email: wollner@mathematik.tu-darmstadt.de organization: Fachbereich Mathematik, Technische Universität Darmstadt |
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| Cites_doi | 10.3934/dcds.2000.6.431 10.1137/120885772 10.1007/s10589-011-9410-6 10.1137/S0363012995283637 10.1137/080744608 10.1137/S0363012900382011 10.1007/s00211-017-0906-6 10.1137/070694016 10.1007/s10589-013-9537-8 10.1137/070694028 10.1137/140960645 10.1137/140961778 10.1137/15M103412X 10.1007/s10589-013-9541-z 10.1137/120871687 10.1016/0022-247X(73)90022-X 10.1007/978-0-387-75934-0 10.1007/s10589-005-4559-5 10.1137/100793888 10.1051/m2an/2009046 10.1365/s13291-014-0109-3 10.1007/s00211-013-0562-4 10.1137/07068240X 10.1137/S0363012998345627 10.1007/s00211-011-0409-9 10.3934/dcdsb.2012.17.1473 10.1137/0331045 10.1007/BF01396760 10.4208/jcm.1006-m3213 |
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| DOI | 10.1007/s10957-018-1311-8 |
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| Keywords | 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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| SubjectTerms | 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 |
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| Title | A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems |
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