Numerical examples

We consider elliptic problems with the following specifications: the cost functional is of tracking type (2.13), the elliptic operator in (2.2) is the Laplacian $\,A=-\Delta\,$ on the unit square $\, \Omega = (0,1) \times (0,1)\,,$ and the control and state constraints are box constraints given in (2.14). The choice of symmetric functions $\,y_d(x)\,$ and $\, u_d(x)\,$ in the tracking functional, implies that the optimal control is the same on every edge of $\,\Gamma = \partial \Omega\,.$ However, we have treated the discretized controls on every edge of $\,\Gamma\,$ as independent optimization variables. The symmetry of the optimal control will then be a result of the optimization procedure.

Tests for the following examples were run with different stepsizes and starting values. For convenience, in the sequel we shall report on results obtained for fixed stepsize and starting values:

$$N=99\,, \; h=1/(N+1)=1/100\,, \quad u_{ij} = 0, \; y_{ij} = 0 \,.$$

First, we report on 4 examples with Dirichlet boundary conditions for which partial numerical results are available in the literature.