Error estimates for the Lavrentiev regularization of elliptic optimal control problems

We focus on optimal control problems governed by partial differential equations. The presence of constraints on the state provides numerical and analytical difficulties. We treat these obstacles introducing a Lavrentiev regularization. The key issue is addressed to the analytical investigation of the convergence order of solutions based on certain source representation arguments. A virtual control approach is investigated for problems with additional control constraints. Numerical experiments are presented.

[1]  M. M. Lavrentiev.  Formulation of some Improperly Posed Problems of Mathematical Physics , 1967 .

[2]  Fredi Tröltzsch,et al.  Optimal Control of PDEs with Regularized Pointwise State Constraints , 2006, Comput. Optim. Appl..

[3]  Arnd Rösch,et al.  On the regularization error of state constrained Neumann control problems , 2008 .

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

[5]  Arnd Rösch,et al.  Primal-Dual Active Set Strategy for a General Class of Constrained Optimal Control Problems , 2002, SIAM J. Optim..

[6]  Kazufumi Ito,et al.  Semi-smooth Newton methods for state-constrained optimal control problems , 2003, Syst. Control. Lett..

[7]  Arnd Rösch,et al.  Error Estimates for the Regularization of Optimal Control Problems with Pointwise Control and State Constraints , 2008 .

[8]  Fredi Tröltzsch,et al.  Existence of Regular Lagrange Multipliers for a Nonlinear Elliptic Optimal Control Problem with Pointwise Control-State Constraints , 2006, SIAM J. Control. Optim..

[9]  Fredi Tröltzsch,et al.  On two numerical methods for state-constrained elliptic control problems , 2007, Optim. Methods Softw..

[10]  Ulrich Tautenhahn,et al.  Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions , 2004 .

[11]  C. Meyer.  Optimal control of semilinear elliptic equations with applications to sublimation crystal growth , 2006 .

[12]  Martin Weiser,et al.  Function space interior point methods for PDE constrained optimization , 2004 .

[13]  E. Casas.  Boundary control of semilinear elliptic equations with pointwise state constraints , 1993 .

[14]  Jaan Janno.  Lavrent'ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation , 2000 .

[15]  Karl Kunisch,et al.  Stationary optimal control problems with pointwise state constraints , 2006 .

[16]  C. Meyer.  Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints , 2008 .

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

[18]  Michael Hinze,et al.  Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems , 2010, Comput. Optim. Appl..

[19]  Michael Ulbrich,et al.  Semismooth Newton Methods for Operator Equations in Function Spaces , 2002, SIAM J. Optim..

[20]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

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

[22]  Fredi Tröltzsch,et al.  The convergence of an interior point method for an elliptic control problem with mixed control-state constraints , 2008, Comput. Optim. Appl..

[23]  Arnd Rösch,et al.  Lipschitz stability for elliptic optimal control problems with mixed control-state constraints , 2010 .

[24]  Eduardo Casas,et al.  Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints , 2008 .