In this paper we study Lavrentiev-type regularization concepts for linear-quadratic parabolic control problems with pointwise state constraints. In the first part, we apply classical Lavrentiev regularization to a problem with distributed control, whereas in the second part, a Lavrentiev-type regularization method based on the adjoint operator is applied to boundary control problems with state constraints in the whole domain. The analysis for both classes of control problems is investigated and numerical tests are conducted. Moreover the method is compared with other numerical techniques.
Mots clés : optimal control, parabolic equation, pointwise state constraints, boundary control, Lavrentiev-type regularization
@article{COCV_2009__15_2_426_0,
author = {Neitzel, Ira and Tr\"oltzsch, Fredi},
title = {On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {426--453},
publisher = {EDP-Sciences},
volume = {15},
number = {2},
year = {2009},
doi = {10.1051/cocv:2008038},
mrnumber = {2513093},
zbl = {1171.49017},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008038/}
}
TY - JOUR AU - Neitzel, Ira AU - Tröltzsch, Fredi TI - On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 426 EP - 453 VL - 15 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008038/ DO - 10.1051/cocv:2008038 LA - en ID - COCV_2009__15_2_426_0 ER -
%0 Journal Article %A Neitzel, Ira %A Tröltzsch, Fredi %T On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 426-453 %V 15 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008038/ %R 10.1051/cocv:2008038 %G en %F COCV_2009__15_2_426_0
Neitzel, Ira; Tröltzsch, Fredi. On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 426-453. doi : 10.1051/cocv:2008038. http://www.numdam.org/articles/10.1051/cocv:2008038/
[1] and , Optimal control problems with mixed control-state constraints. SIAM J. Control 39 (2000) 1391-1407. | MR | Zbl
[2] , and , Asymptotic analysis of some control problems. Asymptotic Anal. 24 (2000) 343-366. | MR | Zbl
[3] and , Primal-dual active set strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22 (2002) 193-224. | MR | Zbl
[4] , and , Primal-dual strategy for constrained optimal control problems. SIAM J. Control Opt. 37 (1999) 1176-1194. | MR | Zbl
[5] , , and , A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11 (2000) 495-521. | MR | Zbl
[6] and , Discretize Then Optimize. Technical Report M&CT-TECH-03-01, Phantom Works, Mathematics & Computing Technology. A Division of The Boeing Company (2003).
[7] and , Discretize then Optimize, in Mathematics in Industry: Challenges and Frontiers A Process View: Practice and Theory, D.R. Ferguson and T.J. Peters Eds., SIAM Publications, Philadelphia (2005). | MR
[8] , and , Direct transcription solution of optimal control problems with higher order state constraints: theory vs. practice. Optim. Engineering 8 (2007) 1-19. | MR | Zbl
[9] , Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Opt. 31 (1993) 993-1006. | MR | Zbl
[10] , Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Opt. 35 (1997) 1297-1327. | MR | Zbl
[11] and , Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal. 45 (2007) 1937-1953. | MR | Zbl
[12] , and , The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2003) 865-888. | MR | Zbl
[13] , and , Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems. Numer. Math. 108 (2008) 571-603. | MR | Zbl
[14] and , Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. Theory Methods Appl. 41 (2000) 591-616. | MR | Zbl
[15] and , Semi-smooth Newton methods for state-constrained optimal control problems. Systems Control Lett. 50 (2003) 221-228. | MR | Zbl
[16] and , Advantages of Nonlinear Programming Based Methodologies for Inequality Path Constrained Optimal Control Problems - An Analysis of the Betts and Campbell Heat Conduction Problem. Technical report, Chemical Engineering Department Carnegie Mellon, University Pittsburgh, USA (2005).
[17] and , Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim. 13 (2002) 321-334. | MR | Zbl
[18] , Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971). | MR | Zbl
[19] and , 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, Avignon, September 20-24, 2004, A. Seeger Ed., Lectures Notes in Economics and Mathematical Systems, Springer-Verlag (2005). | MR
[20] , and , Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33 (2006) 209-228. | MR | Zbl
[21] , and , On two numerical methods for state-constrained elliptic control problems. Optim. Methods Software 22 (2007) 871-899. | MR | Zbl
[22] and , On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints. Technical Report 24-03, SPP 1253 (2008). | MR
[23] , and , The convergence of an interior point method for an elliptic control problem with mixed control-state constraints. Comput. Optim. Appl. 39 (2008) 183-218. | MR | Zbl
[24] and , Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. S. 6 (2000) 431-450. | MR | Zbl
[25] and , Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39 (1999 ) 143-177. | MR | Zbl
[26] and , Existence of regular Lagrange multipliers for a nonlinear elliptic optimal control problem with pointwise control-state constraints. SIAM J. Control Opt. 45 (2006) 548-564. | MR | Zbl
[27] and , Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim. 17 (2006) 776-794. | MR | Zbl
[28] and , On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control Opt. 46 (2007) 1098-1115. | MR
[29] , The control reduced interior point method. A function space oriented algorithmic approach. Ph.D. thesis, Freie Universität Berlin, Germany (2006).
[30] and , A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints. Comput. Optim. Appl. DOI: 10.1007/s10589-007-9114-0 (2008) online first. | MR | Zbl
Cité par Sources :
