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2006-02

Convergence of a finite element approximation to a state constrained elliptic control problem

by Deckelnick, K., Hinze, M..


Series: 2006-02, Preprints

MSC:
49J20 Optimal control problems involving partial differential equations

Abstract:
We consider an elliptic optimal control problem with pointwise
state constraints. The cost functional is approximated by a
sequence of functionals which are obtained by discretizing
the state equation with the help of linear finite elements
and enforcing the state constraints in the nodes of the
triangulation. The corresponding minima are shown to converge
in L^2 to the exact control as the discretization parameter
tends to zero both in two and three space dimensions.
Furthermore, error bounds both for control and state are
obtained in the two-dimensional case. Finally, we present
numerical examples which confirm our analytical findings.

Keywords:
elliptic optimal control problem, state contraints, error estimates