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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 59, 2010 - Issue 6
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Original Articles

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

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Pages 833-849 | Received 16 Jan 2008, Accepted 10 Feb 2009, Published online: 30 Oct 2009

References

  • Adams , R . 1975 . Sobolev Spaces , New York : Academic Press .
  • Alt , W . 1991 . Local stability of solutions to differentiable optimization problems in Banach spaces . J. Opt. Theory Appl. , 70 : 443 – 466 .
  • Alt , W . 1997 . “ Discretization and mesh-independence of Newton's method for generalized equations ” . In Mathematical Programming with Data Perturbations V, volume 195 of Lecture Notes in Pure and Applied Mathematics , Edited by: Fiacco , AV . 1 – 30 . New York : Marcel Dekker .
  • Alt , W and Malanowski , K . 1993 . The Lagrange–Newton method for non-linear optimal control problems . Comput. Opt. Appl. , 2 : 77 – 100 .
  • Bonnans , F and Shapiro , A . 2000 . Perturbation Analysis of Optimization Problems , Berlin : Springer .
  • Dontchev , AL and Hager , WW . 1994 . Implicit functions, Lipschitz maps, and stability in optimization . Math. Oper. Res. , 19 : 753 – 768 .
  • Dontchev , AL , Hager , WW , Poore , AB and Yang , B . 1995 . Optimality, stability, and convergence in nonlinear control . Applied Math. Opt. , 31 : 297 – 326 .
  • Griesse , R . 2006 . Lipschitz stability of solutions to some state-constrained elliptic optimal control problems . J. Anal. Appl. , 25 : 435 – 455 .
  • Grisvard , P . 1985 . Elliptic Problems in Nonsmooth Domains , Boston : Pitman .
  • Ito , K and Kunisch , K . 1992 . Sensitivity analysis of solutions to optimization problems in Hilbert spaces with applications to optimal control and estimation . J. Diff. Equat. , 99 : 1 – 40 .
  • Kinderlehrer , D and Stampacchia , G . 1980 . An Introduction to Variational Inequalities and Their Applications , New York : Academic Press .
  • Malanowski , K . 2001 . Stability and sensitivity analysis for optimal control problems with controlstate constraints . Dissertationes Mathematicae (Rozprawy Matematyczne), , 394 : 1 – 50 .
  • Malanowski , K , Büskens , C and Maurer , H . 1997 . “ Convergence of approximations to nonlinear optimal control problems ” . In Mathematical Programming with Data Perturbations V, volume 195 of Lecture Notes in Pure and Applied Mathematics , Edited by: Fiacco , AV . 253 – 284 . New York : Marcel Dekker .
  • Malanowski , K and Tröltzsch , F . 2000 . Lipschitz stability of solutions to parametric optimal control for elliptic equations . Control Cybern. , 29 : 237 – 256 .
  • Meyer , C , Prüfert , U and Tröltzsch , F . 2007 . On two numerical methods for state-constrained elliptic control problems . Opt. Meth. Software , 22 : 871 – 899 .
  • Meyer , C , Rösch , A and Tröltzsch , F . 2005 . Optimal control of PDEs with regularized pointwise state constraints . Comput. Opt. Appl. , 33 : 209 – 228 .
  • Meyer , C and Tröltzsch , F . 2006 . “ On an elliptic optimal control problem with pointwise mixed control-state constraints ” . In Recent Advances in Optimization. Proceedings of the 12th French-German-Spanish Conference on Optimization, volume 563 of Lecture Notes in Economics and Mathematical Systems , Edited by: Seeger , A . 187 – 204 . New York : Springer .
  • Robinson , MS . 1980 . Strongly regular generalized equations . Math. Oper. Res. , 5 : 43 – 62 .
  • Rösch , A and Tröltzsch , F . 2006 . Existence of regular Lagrange multipliers for elliptic optimal control problems with pointwise control-state constraints . SIAM J. Contr. Opt. , 45 ( 2 ) : 548 – 564 .
  • Rösch , A and Wachsmuth , D . 2006 . Regularity of solutions for an optimal control problem with mixed control-state constraints . TOP , 14 : 263 – 278 .
  • Zeidler , E . 1995 . Applied Functional Analysis: Main Principles and their Applications , New York : Springer .

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