Second Order Sufficient Conditions and the SQP Method for Optimal Control Problems with Mixed Constraints

Start: 2005-09-01
End: 2008-08-31
Principal Investigators: Roland Herzog, Arnd Rösch
Staff: Nataliya Metla
Funded by: FWF within the Individual Research Grant funding scheme

Project Description

Many technical processes are described by partial differential equations. The optimization of such processes or identification of material parameters leads to optimal control problems for partial differential equations. Naturally, some quantities of the process have to be restricted to admissible ranges. The scope of this project covers optimal control of elliptic and parabolic partial differential equations with pointwise inequality constraints in space and time.

Typically, nonlinear functions are involved in real-life problems. In turn, necessary and sufficient optimality conditions of nonlinear optimal control problems contain first and second derivatives of these nonlinearities. Sufficient optimality conditions can ensure stability under perturbations of the solutions of the investigated optimal control problems. Moreover, they represent the key to prove convergence of fast and efficient numerical methods.

Until now, sufficient optimality conditions, stability results, and convergence of fast numerical methods are only known in case the pointwise inequality constraints affect solely the controls of the system. In contrast, real-life problems contain typically both, pointwise inequality constraints for controls and process quantities, i.e., states. Inequality constraints for process quantities alone lead to mathematical problems which are far from being solved.

In this project, we will establish sufficient optimality conditions and we will prove stability results and convergence of the SQP-method for mixed constrained optimal control problems. Pointwise inequality conditions containing controls and process quantities are simultaneously involved in such constraints. These theory developed in this project will guarantee reliable numerical results for arbitrary fine discretizations of the involved partial differential equations.

Associated Publications

Walter Alt, Roland Griesse , Nataliya Metla and Arnd Rösch

Lipschitz stability for elliptic optimal control problems with mixed control-state constraints

Optimization 59(6), p.833-849, 2010

doi:10.1080/02331930902863749

bibtex

@ARTICLE{AltGriesseMetlaRoesch:2010:1,
  AUTHOR = {Alt, Walter and Griesse, Roland and Metla, Nataliya and Rösch, Arnd},
  DATE = {2010},
  DOI = {10.1080/02331930902863749},
  JOURNALTITLE = {Optimization},
  NUMBER = {6},
  PAGES = {833--849},
  TITLE = {Lipschitz stability for elliptic optimal control problems with mixed control-state constraints},
  VOLUME = {59},
}

Roland Griesse , Nataliya Metla and Arnd Rösch

Local quadratic convergence of SQP for elliptic optimal control problems with mixed control-state constraints

Control and Cybernetics 39(3), p.717-738, 2010

bibtex

@ARTICLE{GriesseMetlaRoesch:2010:1,
  AUTHOR = {Griesse, Roland and Metla, Nataliya and Rösch, Arnd},
  DATE = {2010},
  JOURNALTITLE = {Control and Cybernetics},
  NUMBER = {3},
  PAGES = {717--738},
  TITLE = {Local quadratic convergence of SQP for elliptic optimal control problems with mixed control-state constraints},
  VOLUME = {39},
}

Roland Griesse and Daniel Wachsmuth

Sensitivity analysis and the adjoint update strategy for optimal control problems with mixed control-state constraints

Computational Optimization and Applications 44(1), p.57-81, 2009

doi:10.1007/s10589-008-9181-x

bibtex

@ARTICLE{GriesseWachsmuth:2009:1,
  AUTHOR = {Griesse, Roland and Wachsmuth, Daniel},
  DATE = {2009},
  DOI = {10.1007/s10589-008-9181-x},
  JOURNALTITLE = {Computational Optimization and Applications},
  NUMBER = {1},
  PAGES = {57--81},
  TITLE = {Sensitivity analysis and the adjoint update strategy for optimal control problems with mixed control-state constraints},
  VOLUME = {44},
}

Roland Griesse , Nataliya Metla and Arnd Rösch

Convergence analysis of the SQP method for nonlinear mixed-constrained elliptic optimal control problems

Journal of Applied Mathematics and Mechanics 88(10), p.776-792, 2008

doi:10.1002/zamm.200800036

bibtex

@ARTICLE{GriesseMetlaRoesch:2008:2,
  AUTHOR = {Griesse, Roland and Metla, Nataliya and Rösch, Arnd},
  DATE = {2008},
  DOI = {10.1002/zamm.200800036},
  JOURNALTITLE = {Journal of Applied Mathematics and Mechanics},
  NUMBER = {10},
  PAGES = {776--792},
  TITLE = {Convergence analysis of the SQP method for nonlinear mixed-constrained elliptic optimal control problems},
  VOLUME = {88},
}

Roland Griesse

Lipschitz stability of solutions to some state-constrained elliptic optimal control problems

Journal of Analysis and its Applications 25, p.435-455, 2006

doi:10.4171/ZAA/1300

bibtex

@ARTICLE{Griesse:2006:1,
  AUTHOR = {Griesse, Roland},
  DATE = {2006},
  DOI = {10.4171/ZAA/1300},
  JOURNALTITLE = {Journal of Analysis and its Applications},
  PAGES = {435--455},
  TITLE = {Lipschitz stability of solutions to some state-constrained elliptic optimal control problems},
  VOLUME = {25},
}

Arnd Rösch and Fredi Tröltzsch

Existence of regular Lagrange multipliers for a nonlinear elliptic optimal control problem with pointwise control-state constraints

SIAM Journal on Control and Optimization 45(2), p.548-564, 2006

doi:10.1137/050625114

bibtex

@ARTICLE{RoeschTroeltzsch:2006:1,
  AUTHOR = {Rösch, Arnd and Tröltzsch, Fredi},
  DATE = {2006},
  DOI = {10.1137/050625114},
  JOURNALTITLE = {SIAM Journal on Control and Optimization},
  NUMBER = {2},
  PAGES = {548--564},
  TITLE = {Existence of regular Lagrange multipliers for a nonlinear elliptic optimal control problem with pointwise control-state constraints},
  VOLUME = {45},
}

Arnd Rösch and Fredi Tröltzsch

Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints

SIAM Journal on Optimization 17(3), p.776-794, 2006

doi:10.1137/050625850

bibtex

@ARTICLE{RoeschTroeltzsch:2006:2,
  AUTHOR = {Rösch, Arnd and Tröltzsch, Fredi},
  DATE = {2006},
  DOI = {10.1137/050625850},
  JOURNALTITLE = {SIAM Journal on Optimization},
  NUMBER = {3},
  PAGES = {776--794},
  TITLE = {Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints},
  VOLUME = {17},
}