Approximations of Elliptic Optimal Control Problems with Controls Acting on a Lower Dimensional Manifold

In this paper, we study finite element approximations to some elliptic optimal control problems with controls acting on a lower dimensional manifold which can be a point, a curve, or a surface. We use piecewise linear finite elements to approximate state variables, while utilizing the variational discretization to approximate control variables. We derive several a priori error estimates for optimal controls from different cases depending on the dimensions of the computational domain and the manifold where controls act. We ends up with extensive numerical experiments which confirm our theoretical findings.

[1]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[2]  Andreas Günther,et al.  Hamburger Beiträge zur Angewandten Mathematik Finite element approximation of Dirichlet boundary control for elliptic PDEs on two-and three-dimensional curved domains , 2008 .

[3]  Wei Gong,et al.  A Priori Error Analysis for Finite Element Approximation of Parabolic Optimal Control Problems with Pointwise Control , 2014, SIAM J. Control. Optim..

[4]  Eduardo Casas,et al.  Approximation of boundary control problems on curved domains , 2010, 2010 15th International Conference on Methods and Models in Automation and Robotics.

[5]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[6]  A. H. Schatz,et al.  Interior maximum-norm estimates for finite element methods, part II , 1995 .

[7]  Wenbin Liu,et al.  Local A Posteriori Error Estimates for Convex Boundary Control Problems , 2009, SIAM J. Numer. Anal..

[8]  G. Schwarz.  Hodge Decomposition - A Method for Solving Boundary Value Problems , 1995 .

[9]  M. Holst,et al.  The Lichnerowicz equation on compact manifolds with boundary , 2013, 1306.1801.

[10]  H. Beckert,et al.  J. L. Lions and E. Magenes, Non‐Homogeneous Boundary Value Problems and Applications, II. (Die Grundlehren d. Math. Wissenschaften, Bd. 182). XI + 242 S. Berlin/Heidelberg/New York 1972. Springer‐Verlag. Preis geb. DM 58,— , 1973 .

[11]  Eduardo Casas,et al.  Error estimates for the numerical approximation of Neumann control problems , 2008, Comput. Optim. Appl..

[12]  Serge Alinhac,et al.  Pseudo-differential Operators and the Nash-Moser Theorem , 2007 .

[13]  Karl Kunisch,et al.  Approximation of Elliptic Control Problems in Measure Spaces with Sparse Solutions , 2012, SIAM J. Control. Optim..

[14]  Wei Gong,et al.  Mixed Finite Element Method for Dirichlet Boundary Control Problem Governed by Elliptic PDEs , 2011, SIAM J. Control. Optim..

[15]  Alexander Khapalov,et al.  Mobile Point Controls Versus Locally Distributed Ones for the Controllability of the Semilinear Parabolic Equation , 2001, SIAM J. Control. Optim..

[16]  M. Chipot.  Finite Element Methods for Elliptic Problems , 2000 .

[17]  Wenbin Liu,et al.  A Posteriori Error Estimates for Distributed Convex Optimal Control Problems , 2001, Adv. Comput. Math..

[18]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

[19]  Wenbin Liu,et al.  Adaptive Finite Element Methods for Optimal Control Governed by PDEs: C Series in Information and Computational Science 41 , 2008 .

[20]  Jean-Pierre Raymond,et al.  CONTROL PROBLEMS FOR CONVECTION-DIFFUSION EQUATIONS WITH CONTROL LOCALIZED ON MANIFOLDS , 2001 .

[21]  Michael Hinze,et al.  A note on variational discretization of elliptic Neumann boundary control , 2009 .

[22]  Boris Vexler,et al.  A Priori Error Analysis for Discretization of Sparse Elliptic Optimal Control Problems in Measure Space , 2013, SIAM J. Control. Optim..

[23]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[24]  James H. Bramble,et al.  A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries , 1994 .

[25]  Carlos Castro,et al.  Unique Continuation and Control for the Heat Equation from an Oscillating Lower Dimensional Manifold , 2004, SIAM J. Control. Optim..

[26]  Jean-Pierre Raymond,et al.  Control localized on thin structures for semilinear parabolic equations , 2002 .

[27]  M. Hinze,et al.  Hamburger Beiträge zur Angewandten Mathematik Optimal Control of the Laplace-Beltrami operator on compact surfaces-concept and numerical treatment , 2011 .

[28]  E. Casas.  L2 estimates for the finite element method for the Dirichlet problem with singular data , 1985 .

[29]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[30]  Richard S. Falk,et al.  Approximation of a class of optimal control problems with order of convergence estimates , 1973 .

[31]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[32]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

[33]  Jean-Pierre Raymond,et al.  ESTIMATES FOR THE NUMERICAL APPROXIMATION OF DIRICHLET BOUNDARY CONTROL FOR SEMILINEAR ELLIPTIC EQUATIONS , 2006 .

[34]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[35]  Boris Vexler,et al.  Optimal A Priori Error Estimates of Parabolic Optimal Control Problems with Pointwise Control , 2013, SIAM J. Numer. Anal..

[36]  Carlos E. Kenig,et al.  The Inhomogeneous Dirichlet Problem in Lipschitz Domains , 1995 .

[37]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .