Numerical analysis for the approximation of optimal control problems with pointwise observations

In this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.

[1]  K. Kunisch,et al.  Primal-Dual Strategy for Constrained Optimal Control Problems , 1999 .

[2]  Michael Hinze,et al.  Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem , 2007, SIAM J. Numer. Anal..

[3]  T. Geveci,et al.  On the approximation of the solution of an optimal control problem governed by an elliptic equation , 1979 .

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

[5]  Pablo Gamallo,et al.  Finite Element Methods in Local Active Control of Sound , 2004, SIAM J. Control. Optim..

[6]  Rodolfo Rodríguez,et al.  A posteriori error estimates for elliptic problems with Dirac delta source terms , 2006, Numerische Mathematik.

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

[8]  Ricardo H. Nochetto,et al.  Pointwise a posteriori error estimates for elliptic problems on highly graded meshes , 1995 .

[9]  W. Han,et al.  A numerical method for a Cauchy problem for elliptic partial differential equations , 2007 .

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

[11]  Wenbin Liu,et al.  A Posteriori Error Estimates for Control Problems Governed by Stokes Equations , 2002, SIAM J. Numer. Anal..

[12]  Jianxin Zhou,et al.  Constrained LQR Problems in Elliptic Distributed Control Systems with Point Observations , 1996 .

[13]  Jianxin Zhou,et al.  Constrained LQR Problems in Elliptic Distributed Control Systems with Point Observations--- on Convergence Rates , 1997 .

[14]  Arnd Rösch,et al.  Superconvergence Properties of Optimal Control Problems , 2004, SIAM J. Control. Optim..

[15]  E. Casas.  Control of an elliptic problem with pointwise state constraints , 1986 .

[16]  Rolf Rannacher,et al.  A Priori Error Estimates for the Finite Element Discretization of Elliptic Parameter Identification Problems with Pointwise Measurements , 2005, SIAM J. Control. Optim..

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