A morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1181-1206.

We design and analyze a Morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints on convex polygonal domains. It is based on the formulation of the optimal control problem as a fourth order variational inequality. Numerical results that illustrate the performance of the method are also presented.

DOI : 10.1051/cocv/2017031
Classification : 49J20, 65K15, 65N30
Mots clés : Elliptic distributed optimal control problem, pointwise state and control constraints, fourth order variational inequality, Morley element
Brenner, Susanne C. 1 ; Gudi, Thirupathi 1 ; Porwal, Kamana 1 ; Sung, Li-yeng 1

1
@article{COCV_2018__24_3_1181_0,
     author = {Brenner, Susanne C. and Gudi, Thirupathi and Porwal, Kamana and Sung, Li-yeng},
     title = {A morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1181--1206},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {3},
     year = {2018},
     doi = {10.1051/cocv/2017031},
     mrnumber = {3877198},
     zbl = {1412.49025},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2017031/}
}
TY  - JOUR
AU  - Brenner, Susanne C.
AU  - Gudi, Thirupathi
AU  - Porwal, Kamana
AU  - Sung, Li-yeng
TI  - A morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2018
SP  - 1181
EP  - 1206
VL  - 24
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2017031/
DO  - 10.1051/cocv/2017031
LA  - en
ID  - COCV_2018__24_3_1181_0
ER  - 
%0 Journal Article
%A Brenner, Susanne C.
%A Gudi, Thirupathi
%A Porwal, Kamana
%A Sung, Li-yeng
%T A morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2018
%P 1181-1206
%V 24
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2017031/
%R 10.1051/cocv/2017031
%G en
%F COCV_2018__24_3_1181_0
Brenner, Susanne C.; Gudi, Thirupathi; Porwal, Kamana; Sung, Li-yeng. A morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 1181-1206. doi : 10.1051/cocv/2017031. http://www.numdam.org/articles/10.1051/cocv/2017031/

[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces (2nd Edition). Academic Press, Amsterdam (2003) | MR | Zbl

[2] Th. Apel, A.-M. Sändig and J.R. Whiteman, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 19 (1996) 63–85 | DOI | MR | Zbl

[3] M. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37 (1999) 1176–1194 | DOI | MR | Zbl

[4] M. Bergounioux and K. Kunisch, Primal-dual strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22 (2002) 193–224 | DOI | MR | Zbl

[5] M. Bergounioux and K. Kunisch, On the structure of Lagrange multipliers for state-constrained optimal control problems. Systems Control Lett. 48 (2003) 169–176 | DOI | MR | Zbl

[6] J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 113–124 | DOI | MR | Zbl

[7] J.J. Brannick, H. Li and L.T. Zikatanov, Uniform convergence of the multigrid V-cycle on graded meshes for corner singularities. Numer. Linear Algebra Appl. 15 (2008) 291–306 | DOI | MR | Zbl

[8] S.C. Brenner, C.B. Davis and L.-Y. Sung, A partition of unity method for a class of fourth order elliptic variational inequalities. Comput. Methods Appl. Mech. Engrg. 276 (2014) 612–626 | DOI | MR | Zbl

[9] S.C. Brenner, M. Neilan, A. Reiser and L.-Y. Sung, A C0 interior penalty method for a von Kármán plate. Numer. Math. 135 (2017) 803–832 | DOI | MR | Zbl

[10] S.C. Brenner, M. Oh, S. Pollock, K. Porwal, M. Schedensack and N. Sharma, A C0 interior penalty method for elliptic distributed optimal control problems in three dimensions with pointwise state constraints. In Topics in Numerical Partial Differential Equations and Scientific Computing, edited by S.C. Brenner. vol. 160 of The IMA Volumes in Mathematics and its Applications, Cham-Heidelberg-New York-Dordrecht-London (2016) 1–22. Springer | DOI | MR

[11] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (3rd Edition). Springer Verlag, New York (2008) | DOI | MR

[12] S.C. Brenner and L.-Y. Sung, A new convergence analysis of finite element methods for elliptic distributed optimal control problems with pointwise state constraints. SICON, to appear. | MR

[13] S.C. Brenner, L.-Y. Sung and Y. Zhang, A quadratic C0 interior penalty method for an elliptic optimal control problem with state constraints. In Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, edited by O. Karakashian, X. Feng and Y. Xing. vol. 157 of The IMA Volumes in Mathematics and its Applications (2012 John H. Barrett Memorial Lectures). Cham-Heidelberg-New York-Dordrecht-London. Springer 2013 97–132 | DOI | MR | Zbl

[14] S.C. Brenner, L.-Y. Sung and Y. Zhang, Post-processing procedures for a quadratic C0 interior penalty method for elliptic distributed optimal control problems with pointwise state constraints. Appl. Numer. Math. 95(2015) 99–117 | DOI | MR

[15] S.C.Brenner, K. Wang and J. Zhao, Poincaré-Friedrichs inequalities for piecewise H2 functions. Numer. Funct. Anal. Optim. 25 (2004) 463–478 | DOI | MR | Zbl

[16] E. CasasL2 estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math. 47 (1985) 627–632 | DOI | MR | Zbl

[17] E. Casas, M. Mateos and B. Vexler, New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM: COCV 20 (2014) 803–822 | Numdam | MR | Zbl

[18] S. Cherednichenko and A. Rösch, Error estimates for the regularization of optimal control problems with pointwise control and state constraints. Z. Anal. Anwend. 27 (2008) 195–212 | DOI | MR | Zbl

[19] S. Cherednichenko and A. Rösch, Error estimates for the discretization of elliptic control problems with pointwise control and state constraints. Comput. Optim. Appl. 44 (2009) 27–55 | DOI | MR | Zbl

[20] P.G. Ciarlet, Sur l’élément de Clough et Tocher. RAIRO Anal. Numér. 8 (1974) 19–27 | Numdam | MR | Zbl

[21] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-, Amsterdam (1978) | MR | Zbl

[22] M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Lect. Notes Math. Springer Verlag, Berlin Heidelberg 1341 (1988) | MR | Zbl

[23] T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34 (1980) 441–463 | DOI | MR | Zbl

[24] I. Ekeland and R. Témam, Convex Analysis and Variational Problems. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999) | MR | Zbl

[25] L.C. Evans, Partial Differential Equations (Second Edition). Amer. Math. Soc., Providence, RI (2010) | MR

[26] J. Frehse, On the regularity of the solution of the biharmonic variational inequality. Manuscripta Math. 9 (1973) 91–103 | DOI | MR | Zbl

[27] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer Verlag, Berlin (2001) | MR | Zbl

[28] W. Gong and N. Yan, A mixed finite element scheme for optimal control problems with pointwise state constraints. J. Sci. Comput. 46 (2011) 182–203 | DOI | MR | Zbl

[29] P. Grisvard, Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985) | MR | Zbl

[30] P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris (1992) | MR | Zbl

[31] 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 | DOI | MR | Zbl

[32] L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Springer Verlag, Berlin (1985) | MR | Zbl

[33] K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2008) | DOI | MR | Zbl

[34] D. Kinderlehrer and G. Stampacchia. An Introduction to Variational Inequalities and Their Applications. Society for Industrial and Applied Mathematics, Philadelphia (2000) | DOI | MR | Zbl

[35] W. Liu, W. Gong and N. Yan, A new finite element approximation of a state-constrained optimal control problem. J. Comput. Math. 27(2009) 97–114 | MR | Zbl

[36] D.G. Luenberger, Optimization by Vector Space Methods. John Wiley and Sons Inc., New York (1969) | MR | Zbl

[37] C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybernet. 37 (2008) 51–83 | MR | Zbl

[38] L.S.D. Morley, The triangular equilibrium problem in the solution of plate bending problems. Aero. Quart. 19 (1968) 149–169 | DOI

[39] A. Rösch and D. Wachsmuth, A posteriori error estimates for optimal control problems with state and control constraints. Numer. Math. 120 (2012) 733–762 | DOI | MR | Zbl

Cité par Sources :