The SQP method for control constrained optimal control of the Burgers equation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 649-674.

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.

Classification : 49J20, 49K20, 65Kxx
Mots-clés : Burgers' equation, SQP methods, generalized Newton's method, primal-dual methods, active set strategy 
@article{COCV_2001__6__649_0,
     author = {Tr\"oltzsch, Fredi and Volkwein, Stefan},
     title = {The {SQP} method for control constrained optimal control of the {Burgers} equation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {649--674},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
     mrnumber = {1872392},
     zbl = {1001.49035},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2001__6__649_0/}
}
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
SP  - 649
EP  - 674
VL  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/COCV_2001__6__649_0/
LA  - en
ID  - COCV_2001__6__649_0
ER  - 
%0 Journal Article
%A Tröltzsch, Fredi
%A Volkwein, Stefan
%T The SQP method for control constrained optimal control of the Burgers equation
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2001
%P 649-674
%V 6
%I EDP-Sciences
%U http://www.numdam.org/item/COCV_2001__6__649_0/
%G en
%F COCV_2001__6__649_0
Tröltzsch, Fredi; Volkwein, Stefan. The SQP method for control constrained optimal control of the Burgers equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 649-674. http://www.numdam.org/item/COCV_2001__6__649_0/

[1] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

[2] W. Alt, The Lagrange-Newton method for infinite-dimensional optimization problems. Numer. Funct. Anal. Optim. 11 (1990) 201-224. | Zbl

[3] M. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 35 (1997) 1524-1543. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[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). | Zbl

[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).

[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. | MR | Zbl

[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).

[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. | MR | Zbl

[14] S.M. Robinson, Strongly regular generalized equations. Math. Oper. Res. 5 (1980) 43-62. | MR | Zbl

[15] R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam, Stud. Math. Appl. (1979). | Zbl

[16] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, Appl. Math. Sci. 68 (1988). | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[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] S. Volkwein, Augmented Lagrangian-SQP techniques and optimal control problems for the stationary Burgers equation. Comput. Optim. Appl. 16 (2000) 57-81. | MR | Zbl

[21] S. Volkwein, Distributed control problems for the Burgers equation. Comput. Optim. Appl. 18 (2001) 133-158. | MR | Zbl

[22] S. Volkwein, Optimal control of a phase-field model using the proper orthogonal decomposition. Z. Angew. Math. Mech. 81 (2001) 83-97. | MR | Zbl