Elliptic optimal control problems with L1-control cost and applications for the placement of control devices

Research output: Contribution to journalArticle

Abstract

Elliptic optimal control problems with L1-control cost are analyzed. Due to the nonsmooth objective functional the optimal controls are identically zero on large parts of the control domain. For applications, in which one cannot put control devices (or actuators) all over the control domain, this provides information about where it is most efficient to put them. We analyze structural properties of L1-control cost solutions. For solving the non-differentiable optimal control problem we propose a semismooth Newton method that can be stated and analyzed in function space and converges locally with a superlinear rate. Numerical tests on model problems show the usefulness of the approach for the location of control devices and the efficiency of our algorithm.

Original languageEnglish (US)
Pages (from-to)159-181
Number of pages23
JournalComputational Optimization and Applications
Volume44
Issue number2
DOIs
StatePublished - Nov 2009

Fingerprint

Placement
Optimal Control Problem
Costs
Semismooth Newton Method
Function Space
Structural Properties
Actuator
Optimal Control
Newton-Raphson method
Converge
Structural properties
Zero
Actuators

Keywords

  • Active set method
  • Nonsmooth regularization
  • Optimal actuator location
  • Optimal control
  • Placement of control devices
  • Semismooth Newton

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Control and Optimization

Cite this

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abstract = "Elliptic optimal control problems with L1-control cost are analyzed. Due to the nonsmooth objective functional the optimal controls are identically zero on large parts of the control domain. For applications, in which one cannot put control devices (or actuators) all over the control domain, this provides information about where it is most efficient to put them. We analyze structural properties of L1-control cost solutions. For solving the non-differentiable optimal control problem we propose a semismooth Newton method that can be stated and analyzed in function space and converges locally with a superlinear rate. Numerical tests on model problems show the usefulness of the approach for the location of control devices and the efficiency of our algorithm.",
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AB - Elliptic optimal control problems with L1-control cost are analyzed. Due to the nonsmooth objective functional the optimal controls are identically zero on large parts of the control domain. For applications, in which one cannot put control devices (or actuators) all over the control domain, this provides information about where it is most efficient to put them. We analyze structural properties of L1-control cost solutions. For solving the non-differentiable optimal control problem we propose a semismooth Newton method that can be stated and analyzed in function space and converges locally with a superlinear rate. Numerical tests on model problems show the usefulness of the approach for the location of control devices and the efficiency of our algorithm.

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