by Klaus Deckelnick, Michael Hinze, Tobias Jordan.

**Series:** 2015-11, Preprints

- MSC:
- 49J20 Optimal control problems involving partial differential equations
- 65N12 Stability and convergence of numerical methods

**Abstract:**

We consider an optimal shape design problem for the plate equation, where the variable

thickness of the plate is the design function. This problem can be formulated as a control

in the coefficient PDE-constrained optimal control problem with additional control and

state constraints. The state constraints are treated with Moreau-Yosida regularization of a

dual problem. Variational discretization is employed for discrete approximation of the

optimal control problem. For discretization of the state in the mixed formulation we compare the standard continuous piecewise linear ansatz with a piecewise constant one

based on the lowest-order Raviart-Thomas mixed finite element. We derive bounds for the

discretization and regularization errors and also address the coupling of the

regularization parameter and the finite element grid size. The numerical solution of the

optimal control problem is realized with a semismooth Newton algorithm. Numerical examples

show the performance of the method.

**Keywords:**

elliptic optimal control problem, optimal shape design, pointwise state constraints, Moreau-Yosida regularization, error estimates