A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems
We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial di...
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| Published in: | Numerical functional analysis and optimization Vol. 32; no. 12; pp. 1283 - 1315 |
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| Language: | English |
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Philadelphia, PA
Taylor & Francis Group
01.12.2011
Taylor & Francis |
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| ISSN: | 0163-0563, 1532-2467 |
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| Abstract | We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem. |
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| AbstractList | We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem. |
| Author | Slawig, Thomas Prüfert, Uwe Neitzel, Ira |
| Author_xml | – sequence: 1 givenname: Ira surname: Neitzel fullname: Neitzel, Ira email: neitzel@ma.tum.de organization: Technische Universität Berlin , Fakultät II-Mathematik und Naturwissenschaften – sequence: 2 givenname: Uwe surname: Prüfert fullname: Prüfert, Uwe organization: Technische Universität Bergakademie Freiberg , ZIK Virtuhcon, Institut für Energieverfahrenstechnik und Chemieingenieurwesen – sequence: 3 givenname: Thomas surname: Slawig fullname: Slawig, Thomas organization: Christian-Albrechts-Universität zu Kiel , Institut für Informatik |
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| Keywords | Numerical linear algebra Initial value problem Partial differential equation Implementation Discretization method Parabolic PDEs Regularization method Finite element method Projection operator Parabolic equation Numerical analysis Elliptic equation Boundary value problem Optimal control Smooth function Optimality condition Software Ill posed problem Regularization Optimal PDE control Smooth projection operator |
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| References | Neitzel I. (CIT0018) 2008; 50 Evans L. C. (CIT0007) 1998 Hinze M. (CIT0013) 2008 Hintermüller M. (CIT0010) 2003; 83 Lions J. L. (CIT0017) 1971 Wloka J. (CIT0023) 1992 Prüfert U. (CIT0019) 2007; 87 Chen C. (CIT0003) 1996; 5 Ladyzhenskaya O. A. (CIT0016) 1968 Borzi A. (CIT0001) 2003; 157 Zeidler E. (CIT0024) 1990 Glowinski R. (CIT0008) 1979; 21 Tröltzsch F. (CIT0022) 2010 AB COMSOL (CIT0006) 2007 Grisvard P. (CIT0009) 1985 Selvadurai A. P. S. (CIT0020) 2000 Chen X. (CIT0004) 1998; 67 Ito K. (CIT0015) 2009; 118 Tröltzsch F. (CIT0021) 2000; 7 Hinze M. (CIT0011) 2005; 30 Chen X. (CIT0005) 2000; 38 |
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| SubjectTerms | Calculus of variations and optimal control Exact sciences and technology Mathematical analysis Mathematics Numerical analysis Numerical analysis in abstract spaces Numerical analysis. Scientific computation Numerical linear algebra Optimal PDE control Parabolic PDEs Partial differential equations Sciences and techniques of general use Smooth projection operator |
| Title | A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems |
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