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.

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.

DOI : https://doi.org/10.1051/cocv:2008038
Classification : 49K20,  49N10,  49M05
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},
     zbl = {1171.49017},
     mrnumber = {2513093},
     language = {en},
     url = {www.numdam.org/item/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/item/COCV_2009__15_2_426_0/

[1] N. Arada and J.P. Raymond, Optimal control problems with mixed control-state constraints. SIAM J. Control 39 (2000) 1391-1407. | MR 1825584 | Zbl 0993.49022

[2] N. Arada, H. El Fekih and J.-P. Raymond, Asymptotic analysis of some control problems. Asymptotic Anal. 24 (2000) 343-366. | MR 1797776 | Zbl 0979.49020

[3] M. Bergounioux and K. Kunisch, Primal-dual active set strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22 (2002) 193-224. | MR 1911062 | Zbl 1015.49026

[4] M. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems. SIAM J. Control Opt. 37 (1999) 1176-1194. | MR 1691937 | Zbl 0937.49017

[5] M. Bergounioux, M. Haddou, M. Hintermüller and K. Kunisch, 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 1787272 | Zbl 1001.49034

[6] J.T. Betts and S.L. Campbell, Discretize Then Optimize. Technical Report M&CT-TECH-03-01, Phantom Works, Mathematics & Computing Technology. A Division of The Boeing Company (2003).

[7] J.T. Betts and S.L. Campbell, 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 2333222 | Zbl pre05238864

[8] J.T. Betts, S.L. Campbell and A. Englesone, Direct transcription solution of optimal control problems with higher order state constraints: theory vs. practice. Optim. Engineering 8 (2007) 1-19. | MR 2330464 | Zbl 1170.49311

[9] E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Opt. 31 (1993) 993-1006. | MR 1227543 | Zbl 0798.49020

[10] E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Opt. 35 (1997) 1297-1327. | MR 1453300 | Zbl 0893.49017

[11] K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal. 45 (2007) 1937-1953. | MR 2346365 | Zbl 1154.65055

[12] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13 (2003) 865-888. | MR 1972219 | Zbl 1080.90074

[13] M. Hintermüller, F. Tröltzsch and I. Yousept, Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems. Numer. Math. 108 (2008) 571-603. | MR 2369205 | Zbl 1143.65051

[14] K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. Theory Methods Appl. 41 (2000) 591-616. | MR 1780634 | Zbl 0971.49014

[15] K. Ito and K. Kunisch, Semi-smooth Newton methods for state-constrained optimal control problems. Systems Control Lett. 50 (2003) 221-228. | MR 2010814 | Zbl 1157.49311

[16] S. Kameswaran and L.T. Biegler, 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] K. Kunisch and A. Rösch, Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim. 13 (2002) 321-334. | MR 1951024 | Zbl 1028.49027

[18] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971). | MR 271512 | Zbl 0203.09001

[19] C. Meyer and F. Tröltzsch, 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 2191158 | Zbl pre05070324

[20] C. Meyer, A. Rösch and F. Tröltzsch, Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33 (2006) 209-228. | MR 2208817 | Zbl 1103.90072

[21] C. Meyer, U. Prüfert and F. Tröltzsch, On two numerical methods for state-constrained elliptic control problems. Optim. Methods Software 22 (2007) 871-899. | MR 2360802 | Zbl 1172.49022

[22] I. Neitzel and F. Tröltzsch, On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints. Technical Report 24-03, SPP 1253 (2008). | MR 2536485

[23] U. Prüfert, F. Tröltzsch and M. Weiser, 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 2373245 | Zbl 1144.90511

[24] J.-P. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. S. 6 (2000) 431-450. | MR 1739375 | Zbl 1010.49015

[25] J.-P. Raymond and H. Zidani, Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39 (1999 ) 143-177. | MR 1665668 | Zbl 0922.49013

[26] A. Rösch and F. Tröltzsch, 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 2246090 | Zbl 1108.49019

[27] A. Rösch and F. Tröltzsch, Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim. 17 (2006) 776-794. | MR 2257208 | Zbl 1119.49022

[28] A. Rösch and F. Tröltzsch, 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 2338440 | Zbl pre05288517

[29] A. Schiela, The control reduced interior point method. A function space oriented algorithmic approach. Ph.D. thesis, Freie Universität Berlin, Germany (2006).

[30] F. Tröltzsch and I. Yousept, 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 2471186 | Zbl 1153.65343