Optimization techniques for solving elliptic control problems with control and state constraints. Part 2: Distributed control

Helmut Maurer, Hans Mittelmann

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local Pontryagin minimum principle. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. The problems are formulated as AMPL (R. Fourer, D.M. Gay, and B.W. Kernighan, "AMPL: A modeling Language for Mathematical Programming", Duxbury Press, Brooks-Cole Publishing Company, 1993) scripts and several optimization codes are applied. In particular, it is shown that a recently developed interior point method is able to solve theses problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang controls. The necessary conditions of optimality of checked numerically in the presence of active control and state constraints.

Original languageEnglish (US)
Pages (from-to)141-160
Number of pages20
JournalComputational Optimization and Applications
Volume18
Issue number2
DOIs
StatePublished - Feb 2001

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Control Constraints
State Constraints
Distributed Control
Elliptic Problems
Optimization Techniques
Necessary Conditions of Optimality
Control Problem
Bang-bang Control
Minimum Principle
Interior Point Method
Discretization Scheme
Active Control
Modeling Language
Neumann Boundary Conditions
Mathematical Programming
Nonlinear Programming
Dirichlet Boundary Conditions
Dirichlet
Continue
Discretization

Keywords

  • Control and state constraints
  • Discretization techniques
  • Distributed control
  • Elliptic control problems
  • NLP-methods

ASJC Scopus subject areas

  • Applied Mathematics
  • Control and Optimization
  • Management Science and Operations Research
  • Computational Mathematics

Cite this

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abstract = "Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local Pontryagin minimum principle. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. The problems are formulated as AMPL (R. Fourer, D.M. Gay, and B.W. Kernighan, {"}AMPL: A modeling Language for Mathematical Programming{"}, Duxbury Press, Brooks-Cole Publishing Company, 1993) scripts and several optimization codes are applied. In particular, it is shown that a recently developed interior point method is able to solve theses problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang controls. The necessary conditions of optimality of checked numerically in the presence of active control and state constraints.",
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N2 - Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local Pontryagin minimum principle. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. The problems are formulated as AMPL (R. Fourer, D.M. Gay, and B.W. Kernighan, "AMPL: A modeling Language for Mathematical Programming", Duxbury Press, Brooks-Cole Publishing Company, 1993) scripts and several optimization codes are applied. In particular, it is shown that a recently developed interior point method is able to solve theses problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang controls. The necessary conditions of optimality of checked numerically in the presence of active control and state constraints.

AB - Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local Pontryagin minimum principle. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. The problems are formulated as AMPL (R. Fourer, D.M. Gay, and B.W. Kernighan, "AMPL: A modeling Language for Mathematical Programming", Duxbury Press, Brooks-Cole Publishing Company, 1993) scripts and several optimization codes are applied. In particular, it is shown that a recently developed interior point method is able to solve theses problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang controls. The necessary conditions of optimality of checked numerically in the presence of active control and state constraints.

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