**Author(s) :**
Ira Neitzel
,
Uwe Prüfert
,
Thomas Slawig

Preprint series of the Institute of Mathematics, Technische Universität Berlin

Preprint 21-2010

**MSC 2000**

- 49K20 Problems involving partial differential equations

**Abstract :**

We present a smooth, i.e. differentiable regularization of the projection
formula that occurs in constrained parabolic optimal control
problems. We summarize the optimality conditions in function spaces for
unconstrained and control-constrained problems subject to a class of parabolic
partial differential equations. The optimality conditions are then given by
coupled systems
of parabolic PDEs.
For constrained problems, a non-smooth projection operator occurs in the
optimality conditions.
For this projection operator, we present in detail a regularization method based
on smoothed sign, minimum and maximum functions.
For all three cases, i.e (1) the unconstrained problem, (2) the constrained problem
including the projection, and (3) the regularized projection, we
verify that the optimality conditions can be equivalently
expressed by an elliptic boundary value problem in the space-time
domain. For this problem and all three cases we discuss existence and
uniqueness issues.
Motivated by this elliptic problem, we use a simultaneous space-time
discretization for numerical tests. Here we show how
a standard finite element software environment allows to solve the problem and
thus to verify the applicability of this approach
without much implementational effort. We present numerical results for an
example problem.

**Keywords :**
*optimal control*