Goal-oriented adaptivity in control constrained optimal control of partial differential equations

Dual-weighted goal-oriented error estimates for a class of pointwise control constrained optimal control problems for second order elliptic partial differential equations are derived. It is demonstrated that the constraints give rise to a primal-dual weighted error term representing the mismatch in the complementarity system due to discretization. Besides a theoretical foundation, numerical results are presented.

[1]  Michael Hintermüller,et al.  Goal-oriented adaptivity in control constrained optimal control of partial differential equations , 2009, CDC.

[2]  Ruo Li,et al.  Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems , 2002, SIAM J. Control. Optim..

[3]  Michael Hintermüller,et al.  Goal-Oriented Adaptivity in Pointwise State Constrained Optimal Control of Partial Differential Equations , 2010, SIAM J. Control. Optim..

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

[5]  Michael Hintermüller,et al.  AN A POSTERIORI ERROR ANALYSIS OF ADAPTIVE FINITE ELEMENT METHODS FOR DISTRIBUTED ELLIPTIC CONTROL PROBLEMS WITH CONTROL CONSTRAINTS , 2008 .

[6]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[7]  Claes Johnson,et al.  Introduction to Adaptive Methods for Differential Equations , 1995, Acta Numerica.

[8]  Rüdiger Verführt,et al.  A review of a posteriori error estimation and adaptive mesh-refinement techniques , 1996, Advances in numerical mathematics.

[9]  Rolf Rannacher,et al.  Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept , 2000, SIAM J. Control. Optim..

[10]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[11]  Endre Süli,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

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