We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.
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DOI : 10.1051/cocv/2019068
Keywords: Diffusion equation, PDE-constrained optimization, control-constraints, finite element method, error bounds
@article{COCV_2020__26_1_A78_0,
author = {Gudi, Thirupathi and Sau, Ramesh Ch.},
title = {Finite element analysis of the constrained {Dirichlet} boundary control problem governed by the diffusion problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019068},
mrnumber = {4156825},
zbl = {1460.65147},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019068/}
}
TY - JOUR AU - Gudi, Thirupathi AU - Sau, Ramesh Ch. TI - Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019068/ DO - 10.1051/cocv/2019068 LA - en ID - COCV_2020__26_1_A78_0 ER -
%0 Journal Article %A Gudi, Thirupathi %A Sau, Ramesh Ch. %T Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019068/ %R 10.1051/cocv/2019068 %G en %F COCV_2020__26_1_A78_0
Gudi, Thirupathi; Sau, Ramesh Ch. Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 78. doi: 10.1051/cocv/2019068
[1] , and , Finite element error estimates for Neumann boundary control problems on graded meshes. Comput. Optim. Appl. 52 (2012) 3–28. | MR | Zbl | DOI
[2] , , and , Quadratic finite elements with non matching grids for the unilateral boundary contact. Math. Model. Numer. Anal. 47 (2013) 1185–1205. | MR | Zbl | Numdam | DOI
[3] , and , Singular perturbation for the dirichlet boundary control of elliptic problems. ESAIM: M2AN 37 (2003) 833–850. | MR | Zbl | Numdam | DOI
[4] and Quadratic finite element approximation of the Signorini problem. Math. Comp. 241 (2003) 83–104. | MR | Zbl
[5] and , The Mathematical Theory of Finite Element Methods (Third Edition). Springer-Verlag, New York (2008). | MR | DOI
[6] and , Error estimates for linear-quadratic elliptic control problems, in Analysis and Optimization of Differential Systems. Kluwer Academic Publishing, Boston (2003) 89–100. | MR | Zbl | DOI
[7] and , Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45 (2006) 1586–1611. | MR | Zbl | DOI
[8] and , Error estimates for the numerical approximation of Neumann control problems Comput. Optim. Appl. 39 (2008) 265–295. | MR | Zbl | DOI
[9] and , Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations. Comput. Optim. Appl. 52 (2012) 719–756. | MR | Zbl | DOI
[10] , and , Penalization of Dirichlet optimal control problems. ESAIM: COCV 15 (2009) 782–809. | MR | Zbl | Numdam
[11] , and , A framework for the error analysis of discontinuous finite element methods for elliptic optimal control problems and applications to C0 IP methods. Numer. Funct. Anal. Optim. 36 (2015) 1388–1419. | MR | Zbl | DOI
[12] , and , On the finite element approximation of the Dirichlet boundary control problem. Math. Comp. 86 (2017) 1103–1126. | MR | Zbl
[13] , The Finite Element Method for Elliptic Problems. North-, Amsterdam (1978). | MR | Zbl
[14] , and , Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains. SIAM J. Numer. Anal. 48 (2009) 2798–2819. | MR | Zbl
[15] , and , An Error analysis of Discontinuous Finite Element Methods for the Optimal Control problems governed by Stokes equation. Numer. Funct. Anal. Optim. 40 (2019) 421–460. | MR | Zbl | DOI
[16] and , Optimal convergence for discrete variational inequalities modelling Signorini contact in 2D and 3D without additonal assumptions on the unkown contact set. SIAM J. Numer. Anal. 53 (2015) 1488–1507. | MR | Zbl | DOI
[17] , Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44 (1973) 28–47. | MR | Zbl | DOI
[18] , On the approximation of the solution of an optimal control problems governed by an elliptic equation. ESAIM: M2AN 13 (1979) 313–328. | MR | Zbl | Numdam
[19] and , Elliptic control problems with gradient constraints variational discrete versus piecewise constant controls. Comput. Optim. Appl. 40 (2011) 549–566. | MR | Zbl | DOI
[20] , and , Analysis and finite element approximation of optimal control problems for the stationary Navier–Stokes equations with Dirichlet controls. ESAIM: M2AN 25 (1991) 711–748. | MR | Zbl | Numdam | DOI
[21] , and , Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optim. 30 (1992) 167–181. | MR | Zbl | DOI
[22] , A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45–61. | MR | Zbl | DOI
[23] , , and , Optimization with PDE Constraints. Springer, New York (2009). | MR | Zbl
[24] , Equations aux Dérivées Partielles Elliptiques non Linéaires. Springer-Verlag, New York (2013). | MR | Zbl | DOI
[25] and , Adaptive Finite Element Methods: Optimal Control Governed by PDEs. Beijing Science Press, Beijing (2008).
[26] , and , Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems. SIAM J. Control Optim. 51 (2013) 2585–2611. | MR | Zbl | DOI
[27] and , Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43 (2004) 970–985. | MR | Zbl | DOI
[28] , and , An energy space finite element approach for elliptic Dirichlet boundary control problems Numer. Math. 129 (2014) 723–748. | MR | Zbl | DOI
[29] J.-P. Raymond, Optimal Control of Partial Differential Equations (lecture notes).
[30] , Optimale Steuerung Partieller Differentialgleichungen, 1st ed. Vieweg. Cambridge University Press (2005). | DOI
Cité par Sources :
The authors acknowledge the support from the UGC center for Advanced Study-II. The first author also thank the support from DST MATRICS Grant.
