Example 5.1: To enable a comparison, we consider first the example in Bergounioux, Kunisch , Section 5.2, with the following data:
The following results were obtained, they are explained below:
Cost functional : , CPU seconds : 96
The optimal control is shown in Figure 1. The optimal state and adjoint variable are not shown here, because they are very similar to those in Figure 3 where is replaced by . It should be noted that the output of Lagrange multipliers is provided by AMPL. It is no difficulty for primal-dual methods to produce multipliers since, in fact they are variables that are computed simultaneously with the primal problem variables. The control constraints are not active while the state variable attains its upper bound only in the center of the unit square with dual variable . This is in agreement with the results in . It can be checked that the adjoint equation (3.6) holds in the discretized version with denoting the discretized Laplacian,
This example will also be used to illustrate the numerical process in more detail.
In Table 1 we summarize the data: size of grid, number of iterations of LOQO, times in seconds for the AMPL compilation and the solution by LOQO, the accuracy as the number of correct significant digits in the objective function value. The primal-dual approach yields upper and lower bounds for this value and thus permits such a statement. In all the following examples this accuracy measure was at least 8 and is therefore not listed. Finally, we list the objective function value and one representative value of both the state and the control variable. While the state variable not far from the active center point has a small error, the control at the midpoints of the edges still varies in the fourth digit. Thus, was chosen for the subsequent calculations. Since the AMPL times were in the range of a few seconds or a few percent of the solution times they will not be given below.
We compare LOQO and other solvers with AMPL interface available to us in Table 2. For not all the codes considered (LANCELOT-A, MINOS-5.5, SNOPT-5.3, BPMPD-2.21; links to all codes in ) were successful. Therefore, results are listed for several coarser grids as well as for the finest grid used with LOQO. Given are CPU times on the same platform used above, but scaled to LOQO's time. MINOS reports one problem as infeasible while SNOPT reports two problems as unbounded. The reason for these messages may also be a "bad starting guess". All solvers were given the same AMPL file and AMPL initializes variables with zero. For all solvers except LOQO and BPMPD exceeded the available memory (256MB). The convex QP code BPMPD solves with increasing efficiency compared to LOQO but this is only possible, because, in fact, Example 5.1 is a convex QP problem. BPMPD is not applicable to most of the other examples solved below. For another comparison see Example 5.7.
Example 5.2: All data are the same as in Example 5.1 except that we choose instead of . According to the optimal control law (3.11) we can expect either a bang-bang or a singular control. We find the following results:
Cost functional : , CPU seconds : 78
The optimal control, adjoint variable and optimal state are shown in Figures 2 and 3. Both the control and state constraints do not become active. Hence, the optimal control is totally singular on . This is in accordance with the control law (3.11) since the normal derivative of the adjoint variable satisfies which follows from the numerical results in view of (4.4). In Figure 3 as in Figures 5 and 7 below the vertical axis is labeled since for these examples both the state and the control are plotted in these figures.
Example 5.3: The data are the same as in Example 5.1 except that we choose more restrictive state and control constraints:
We obtain the following results:
Cost functional : , CPU seconds : 103
The optimal control, state and adjoint variable are shown in Figures 4 and 5. 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 . By inspecting the switching function in Figure 4, we can verify the control law (3.10). The assumption underlying (3.10) obviously holds. Let us check the control condition on the bottom edge of at the points . Observe that according to (4.22) since we have . Then in view of the control law (3.10) takes the form
Example 5.4: The data are those from Example 5.3 but now we choose expecting to obtain a bang-bang control in contrast to the totally singular control in Example 5.2. This expectation is met with the following results:
Cost functional : , CPU seconds : 116
The optimal control shown in Figure 6 is indeed bang-bang. The switching function on the bottom edge of is according to (4.22). Then Figure 6 clearly illustrates the control law (3.11):