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## Neumann boundary conditions

Example 5.5: The first problem has a linear partial differential equation and nonlinear Neumann boundary conditions with data:

The following results were obtained:

Cost functional : , CPU seconds : 494

The optimal control and state are shown in Figures 8 and 9. Note that the sign condition (2.5) holds since . The optimal control is continuous and has two boundary arcs with and one boundary arc with . The junction points with the boundary are the points on the bottom edge of . The control law (2.16) was derived under the assumption which obviously holds in this example. By looking at the adjoint variable in Figure 8, the reader may verify that the control law (2.16) is satisfied:

The active set for the state constraint are the midpoints of the edges. The dual variable for this active inequality constraint is . It can be checked that the adjoint equations (2.7) and (2.8) hold observing the scaling (4.13). As an example, let us test the Neumann boundary condition (2.8) at the active point . Hence, we have to verify the relation which corresponds to the equation . We find und can check indeed the Neumann condition with which agrees with .

Example 5.6: The data for the second Neumann problem are:

According to the optimal control law (2.17) we can expect either a bang-bang or a singular control. We get the following results:

Cost functional : , CPU seconds : 864

The optimal control in Figure 10 is indeed bang-bang. The switching function on as shown in Figure 10 obeys the optimal control law (2.17):

The switching points are approximately . Again, the optimal state displayed in Figure 11 is active at the midpoints of the edges. The dual variable for this active inequality constraint is .

Example 5.7: In this example we choose a nonlinear partial differential equation and linear Neumann boundary conditions:

These equations represent a simplified Ginzburg-Landau model for super-conductivity in the absence of internal magnetic fields with the wave function; cf. Ito, Kunisch [16] and Kunisch, Volkwein [18] where tracking functions and control or state constraints have been considered that are different from those used here. LOQO and AMPL provide the results:

Cost functional : , CPU seconds : 317

The optimal control and state are shown in Figures 12 and 13. The optimal control is continuous and has two boundary arcs with and one boundary arc with . The junction points with the boundary are the points on the bottom edge of . The adjoint variable in Figure 12 shows that the control law (2.16) is satisfied:

The active set for the state constraint comprises the points adjacent to the corners of the domain. The dual variable for this active inequality constraint is . For this problem the QP-code BPMPD is not applicaple. Both MINOS and SNOPT return with error messages as in Example 5.1, while LANCELOT solves the example on grids of and needs respectively respectively times as long as LOQO.

Example 5.8: The data are those from Example 5.7 but now we choose expecting to obtain a bang-bang control. The numerical results are:

Cost functional : , CPU seconds : 570

The optimal control shown in Figure 14 is indeed bang-bang. The switching function on as shown in Figure 14 yields the optimal control law (2.17):

The switching points are approximately . Again, the optimal state displayed in Figure 15 is active at the points adjacent to the corners of the domain. The dual variable for this active inequality constraint is .

Next: Conclusions Up: Numerical examples Previous: Dirichlet boundary conditions
Hans D. Mittelmann
2002-11-25