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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Bibliographic Details
Published in:Numerical functional analysis and optimization Vol. 32; no. 12; pp. 1283 - 1315
Main Authors: Neitzel, Ira, Prüfert, Uwe, Slawig, Thomas
Format: Journal Article
Language:English
Published: Philadelphia, PA Taylor & Francis Group 01.12.2011
Taylor & Francis
Subjects:
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
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
ISSN:0163-0563, 1532-2467
Online Access:Get full text
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Summary: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.
ISSN:0163-0563
1532-2467
DOI:10.1080/01630563.2011.597915