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
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     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},
     zbl = {1001.49035},
     mrnumber = {1872392},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2001__6__649_0/}
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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 450957 | Zbl 0314.46030

[2] W. Alt, The Lagrange-Newton method for infinite-dimensional optimization problems. Numer. Funct. Anal. Optim. 11 (1990) 201-224. | Zbl 0694.49022

[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 1466914 | Zbl 0897.49001

[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 1156075 | Zbl 0755.35001

[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 1421341 | Zbl 0856.65064

[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 1316261 | Zbl 0821.49022

[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 0909.49016

[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 1752557 | Zbl 0992.49023

[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 1025.49022

[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 1384957 | Zbl 0860.49023

[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 1442024 | Zbl 0876.93045

[14] S.M. Robinson, Strongly regular generalized equations. Math. Oper. Res. 5 (1980) 43-62. | MR 561153 | Zbl 0437.90094

[15] R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam, Stud. Math. Appl. (1979). | Zbl 0426.35003

[16] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, Appl. Math. Sci. 68 (1988). | MR 953967 | Zbl 0662.35001

[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 1744965 | Zbl 0954.49017

[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 0954.49018

[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 1761295 | Zbl 0974.49020

[21] S. Volkwein, Distributed control problems for the Burgers equation. Comput. Optim. Appl. 18 (2001) 133-158. | MR 1818917 | Zbl 0976.49001

[22] S. Volkwein, Optimal control of a phase-field model using the proper orthogonal decomposition. Z. Angew. Math. Mech. 81 (2001) 83-97. | MR 1818724 | Zbl 1007.49019