The SQP method for control constrained optimal control of the Burgers equation

Fredi Tröltzsch; Stefan Volkwein

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 649-674
  • ISSN: 1292-8119

Abstract

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A Lagrange–Newton–SQP method is analyzed for the optimal control of the Burgers equation. Distributed controls are given, which are restricted by pointwise lower and upper bounds. The convergence of the method is proved in appropriate Banach spaces. This proof is based on a weak second-order sufficient optimality condition and the theory of Newton methods for generalized equations in Banach spaces. For the numerical realization a primal-dual active set strategy is applied. Numerical examples are included.

How to cite

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Tröltzsch, Fredi, and Volkwein, Stefan. "The SQP method for control constrained optimal control of the Burgers equation." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 649-674. <http://eudml.org/doc/90613>.

@article{Tröltzsch2001,
abstract = {A Lagrange–Newton–SQP method is analyzed for the optimal control of the Burgers equation. Distributed controls are given, which are restricted by pointwise lower and upper bounds. The convergence of the method is proved in appropriate Banach spaces. This proof is based on a weak second-order sufficient optimality condition and the theory of Newton methods for generalized equations in Banach spaces. For the numerical realization a primal-dual active set strategy is applied. Numerical examples are included.},
author = {Tröltzsch, Fredi, Volkwein, Stefan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Burgers’ equation; SQP methods; generalized Newton’s method; primal-dual methods; active set strategy; Burgers equation; generalized Newton method; optimal control},
language = {eng},
pages = {649-674},
publisher = {EDP-Sciences},
title = {The SQP method for control constrained optimal control of the Burgers equation},
url = {http://eudml.org/doc/90613},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Tröltzsch, Fredi
AU - Volkwein, Stefan
TI - The SQP method for control constrained optimal control of the Burgers equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 649
EP - 674
AB - A Lagrange–Newton–SQP method is analyzed for the optimal control of the Burgers equation. Distributed controls are given, which are restricted by pointwise lower and upper bounds. The convergence of the method is proved in appropriate Banach spaces. This proof is based on a weak second-order sufficient optimality condition and the theory of Newton methods for generalized equations in Banach spaces. For the numerical realization a primal-dual active set strategy is applied. Numerical examples are included.
LA - eng
KW - Burgers’ equation; SQP methods; generalized Newton’s method; primal-dual methods; active set strategy; Burgers equation; generalized Newton method; optimal control
UR - http://eudml.org/doc/90613
ER -

References

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  1. [1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl0314.46030MR450957
  2. [2] W. Alt, The Lagrange–Newton method for infinite-dimensional optimization problems. Numer. Funct. Anal. Optim. 11 (1990) 201-224. Zbl0694.49022
  3. [3] M. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 35 (1997) 1524-1543. Zbl0897.49001MR1466914
  4. [4] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5: Evolution Problems I. Springer-Verlag, Berlin (1992). Zbl0755.35001MR1156075
  5. [5] A.L. Dontchev, Local analysis of a Newton-type method based on partial linearization, in Proc. of the AMS-SIAM Summer Seminar in Applied Mathematics on Mathematics and Numerical Analysis: Real Number Algorithms, edited by J. Renegar, M. Shub and S. Smale. AMS, Lectures in Appl. Math. 32 (1996) 295-306. Zbl0856.65064MR1421341
  6. [6] A.L. Dontchev, W.W. Hager, A.B. Poore and B. Yang, Optimality, stability, and convergence in optimal control. Appl. Math. Optim. 31 (1995) 297-326. Zbl0821.49022MR1316261
  7. [7] H. Goldberg and F. Tröltzsch, On the Lagrange–Newton-SQP method for the optimal control of semilinear parabolic equations. Optim. Methods Softw. 8 (1998) 225-247. Zbl0909.49016
  8. [8] M. Heinkenschloss and F. Tröltzsch, Analysis of the Lagrange-SQP-Newton Method for the Control of a Phase-Field Equation. Control Cybernet. 28 (1999) 177-211. Zbl0992.49023MR1752557
  9. [9] M. Hintermüller, A primal-dual active set algorithm for bilaterally control constrained optimal control problems. Spezialforschungsbereich F 003, Optimierung und Kontrolle, Projektbereich Optimierung und Kontrolle, Bericht No. 146 (submitted). Zbl1025.49022
  10. [10] M. Hinze and K. Kunisch, Second order methods for time-dependent fluid flow. Spezialforschungsbereich F 003, Optimierung und Kontrolle, Projektbereich Optimierung und Kontrolle, Bericht No. 165 (submitted). Zbl1012.49026
  11. [11] K. Ito and K. Kunisch, Augmented Lagrangian-SQP-Methods for nonlinear optimal control problems of tracking type. SIAM J. Control Optim. 34 (1996) 874-891. Zbl0860.49023MR1384957
  12. [12] K. Kunisch and A. Rösch, Primal-dual strategy for parabolic optimal control problems. Spezialforschungsbereich F 003, Optimierung und Kontrolle, Projektbereich Optimierung und Kontrolle, Bericht No. 154 (submitted). Zbl1028.49027
  13. [13] H.V. Ly, K.D. Mease and E.S. Titi, Some remarks on distributed and boundary control of the viscous Burgers equation. Numer. Funct. Anal. Optim. 18 (1997) 143-188. Zbl0876.93045MR1442024
  14. [14] S.M. Robinson, Strongly regular generalized equations. Math. Oper. Res. 5 (1980) 43-62. Zbl0437.90094MR561153
  15. [15] R. Temam, Navier–Stokes Equations. North-Holland, Amsterdam, Stud. Math. Appl. (1979). Zbl0426.35003
  16. [16] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, Appl. Math. Sci. 68 (1988). Zbl0662.35001MR953967
  17. [17] F. Tröltzsch, Lipschitz stability of solutions to linear-quadratic parabolic control problems with respect to perturbations. Dynam. Contin. Discrete Impuls. Systems 7 (2000) 289-306. Zbl0954.49017MR1744965
  18. [18] F. Tröltzsch, On the Lagrange–Newton-SQP method for the optimal control of semilinear parabolic equations. SIAM J. Control Optim. 38 (1999) 294-312. Zbl0954.49018
  19. [19] S. Volkwein, Mesh-Independence of an Augmented Lagrangian-SQP Method in Hilbert Spaces and Control Problems for the Burgers Equation, Ph.D. Thesis. Department of Mathematics, Technical University of Berlin (1997). 
  20. [20] S. Volkwein, Augmented Lagrangian-SQP techniques and optimal control problems for the stationary Burgers equation. Comput. Optim. Appl. 16 (2000) 57-81. Zbl0974.49020MR1761295
  21. [21] S. Volkwein, Distributed control problems for the Burgers equation. Comput. Optim. Appl. 18 (2001) 133-158. Zbl0976.49001MR1818917
  22. [22] S. Volkwein, Optimal control of a phase-field model using the proper orthogonal decomposition. Z. Angew. Math. Mech. 81 (2001) 83-97. Zbl1007.49019MR1818724

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