Verification of SSC Conditions

In this section numerical results will be reported for the application of the method outlined in the previous section to two parabolic control problems from [1,9] and a total of ten elliptic control problems from [16]-[19]. The first parabolic problem is particularly interesting because for it in [1] an analytical exact solution is given and the continuous SSC conditions are verified. For the sake of completeness, the specification will be given for each problem. The discretizations used are those defined in section 2 for the parabolic and in [16]-[19] for the elliptic problems.

Problem (P) from section 2 is solved with the following data

$$l=\pi/4,\quad T=1,\quad\alpha=\frac{\sqrt 2}2(e^{2/3}-e^{1/3})$$

$$y_T(x)=(e+e^{-1})\cos x,\quad\alpha_1=0,\quad\alpha_2=1$$

$$a(x)=\cos x,\quad a_y(t)=-e^{-2t},\quad a_u(t)=\frac{\sqrt2}2e^{1/3}$$ (5.1)

$$b(t)=\frac14e^{-4t}-\min\bigg(1,\max\bigg(0,\frac{e^t-e^{1/3}}{e^{2/3}-e^{1/3}}\bigg) \bigg)$$

$$\varphi(y)=y\vert y\vert^3,\quad\beta=1.$$

As is shown in [1] a local optimum for this problem is the pair $(\bar y,\bar u)$,

$$\begin{eqnarray*} &&\bar y(x,t)=e^{-t}\cos x\\ &&\bar u(t)=\min\bigg(1,\max\bigg(0,\frac{e^t-e^{1/3}}{e^{2/3}-e^{1/3}}\bigg)\bigg). \end{eqnarray*}$$

The discretization $(P_h)$ of section 2 was coded in AMPL and solved with LOQO. Afterwards, the SSC were checked as described in the previous section resulting in the minimal eigenvalue $\gamma_h$, cf. (4.3).

Table 1: Solution errors for problem 5.1

$1/h$	$u$-error	$y$-error
50	2.662e-4	4.462e-5
100	7.444e-5	1.208e-5
200	1.884e-5	3.128e-6
350	6.289e-6	1.016e-6

Table 2: Minimal eigenvalue for problem 5.1

$1/h$	$\gamma_h$	$\gamma_h/h^2$
50	6.379e-4	1.595
59	4.781e-4	1.664
70	3.356e-4	1.644
80	2.577e-4	1.649
89	2.135e-4	1.691

In Table 1 the maximum errors over the grid points are listed for the computed control and state functions while in Table 2 the smallest eigenvalue is listed as well as the value scaled by $h^{-2}$. The errors exhibit quadratic convergence while the scaled eigenvalue stays nearly constant with a slightly increasing tendency. These facts justify the application of the technique to other control problems for which no exact solution is known and it also shows which scaling of $\gamma_h$ is appropriate for problem $(P_h)$. This scaling is also suggested by known quadratic convergence estimates for fixed control in $L^{\infty}$ since the differential equation as well as all boundary conditions were discretized with second-order consistency. An argument could be made that the smallness of the eigenvalue may indicate that it is, in fact, negative and the relatively coarse discretization prevents it from exhibiting this fact. This seems highly unlikely in the light of the very uniform and even slightly growing behavior of the scaled value combined with the fact that all the eigenvalues were computed and the smallest one is not considerably smaller than the next ones but they grow only gradually.

As a second parabolic case one from [9] was chosen. The data for (P) are

$$l=1,\quad T=1.58,\quad \alpha=.001$$

$$y_T(x)=.5(1-x^2),\quad \alpha_1=-1,\quad\alpha_2=1$$

$$a(x)=0,\quad a_y(t)=0,\quad a_u(t)=0,$$ (5.2)

$$\mbox{(I)} \quad b(t)=0,\quad \varphi(y)=0,\quad\beta=1$$

$$\mbox{(II)} \quad b(t)=0,\quad\varphi(y)=y^2,\quad\beta=0$$

The case (I) leads to a linear-quadratic control problem and had already been considered in [21]. For both cases just the minimal eigenvalue can be listed without and with the same scaling as in the previous example.

Table 3: Minimal eigenvalue for 5.2-I

$y_h$	$\gamma_h$	$\gamma_h/h^2$
60	5.604e-6	2.02e-2
70	4.215e-6	2.06e-2
80	3.249e-6	2.08e-2
90	2.607e-6	2.11e-2

Table 4: Minimal eigenvalue for 5.2-II

$1/h$	$\gamma_h$	$\gamma_h/h^2$
60	2.498e-6	8.99e-3
70	1.866e-6	9.14e-3
80	1.447e-6	9.26e-3
90	1.155e-6	9.36e-3

For 5.2-II we also include a plot of the optimal control and state in Figure 1.

Figure 1: Example 5.2-II, Optimal control and state

Next, we present the data for the elliptic boundary control problems from [16,19] and the eigenvalues obtained. The domain is the unit square in all cases. The problem (EB) together with (2.1) and (2.2) is considered and $u_d=0$ in all cases.

(EB-1)

$$\begin{eqnarray*} &&d(x,y)=-20,\quad a(x,u)=u,\quad \Gamma_2=\emptyset,\\ &&y_d... ..._2-1),\\ &&\alpha=.01,\quad\psi(x)=3.5,\quad u_1=0,\quad u_2=10 \end{eqnarray*}$$

Table 5: Minimal eigenvalue for (EB-1)

$1/h$	$\gamma_h$	$\gamma_h/h$
60	3.333e-4	2.0e-2
70	2.857e-4	2.0e-2
80	2.500e-4	2.0e-2
90	2.222e-4	2.0e-2

(EB-2) as above, but $\varphi(x)=3.2$, $u_1=1.6$, $u_2=3.2$.

Table 6: Minimal eigenvalue for (EB-2)

$1/h$	$\gamma_h$	$\gamma_h/h$
60	3.333e-4	2.0e-2
70	2.857e-4	2.0e-2
80	2.521e-4	2.0e-2
90	2.222e-4	2.0e-2

A plot of the optimal control and state for (EB-2) is given in Figure 2.

Figure 2: Example (EB-2), Optimal control and state

(EB-3)

$$\begin{eqnarray*}%%\label{EB-1} &&d(x,y)=0,\quad b(x,y,u)=u-y^2,\quad \Gamma_1=\... ...,\\ &&\alpha=.01,\quad\psi(x)=2.071,\quad u_1=3.7,\quad u_2=4.5 \end{eqnarray*}$$

Table 7: Minimal eigenvalue for (EB-3)

$1/h$	$\gamma_h$	$\gamma_h/h$
60	1.534e-4	9.20e-3
70	1.300e-4	9.10e-3
80	1.135e-4	9.08e-3
90	1.008e-4	9.07e-3

(EB-4)

$$\begin{eqnarray*} &&d(x,y)=y-y^3,\quad b(x,y,u)=u,\quad \Gamma_1=\emptyset,\\ &... ...\\ &&\alpha=.01,\quad\varphi(x)=2.7,\quad u_1=1.8,\quad u_2=2.5 \end{eqnarray*}$$

Table 8: Minimal eigenvalue for (EB-4)

$1/h$	$\gamma_h$	$\gamma_h/h$
60	1.654e-4	9.92e-3
70	1.412e-4	9.88e-3
80	1.232e-4	9.86e-3
90	1.094e-4	9.85e-3

(EB-5)

$$\begin{eqnarray*} &&d(x,y)\equiv0;\quad g(x,y,u)\equiv0;\quad b(x,y,u)=0,\quad\m... ...ma_2\\ &&y_d\equiv1,\quad\Omega_0=[.25,.75]^2,\quad \alpha=.005 \end{eqnarray*}$$

Table 9: Minimal eigenvalue for (EB-5)

$1/h$	$\gamma_h$	$\gamma_h/h^2$
40	1.412e-4	2.26e-1
56	7.530e-5	2.36e-1
72	4.618e-5	2.39e-1
88	3.026e-5	2.34e-1

In all cases except (EB-5) which is different from the others in key elements, the scaling of $\gamma_h$ by $h^{-1}$ was appropriate. The following five elliptic distributed control examples were considered in [17,19].

(ED-1)

$$\begin{eqnarray*} &&f(x,y,u)=u^2-.8uy,\quad g(x,y)=0,\\ &&d(x,y,u)=-y(a(x)-u-y)... ...=7+4\sin(2\pi x_1x_2)\\ &&\psi(x)=7.1,\quad u_1=1.7,\quad u_2=2 \end{eqnarray*}$$

Table 10: Minimal eigenvalue for (ED-1)

$1/h$	$\gamma_h$	$\gamma_h/h^2$
40	1.250e-3	2.0e0
50	8.002e-4	2.0e0
60	5.557e-4	2.0e0

A plot of the optimal control and state for (ED-1) is given in Figure 3.

Figure 3: Example (ED-1), Optimal control and state

(ED-2)

$$\begin{eqnarray*} &&d(x,y,u)=-y+y^3-u,\\ &&y_2(x)=0,\quad \Gamma=\Gamma_2,\quad... ...x_1-1)+x_2(x_2-1))\\ &&\psi(x)=1.85,\quad u_1=1.5,\quad u_2=4.5 \end{eqnarray*}$$

Table 11: Minimal eigenvalue for (ED-2)

$1/h$	$\gamma_h$	$\gamma_h/h^2$
40	6.028e-6	9.64e-3
50	3.983e-6	9.96e-3
60	2.719e-6	9.79e-3

(ED-3)

$$\begin{eqnarray*} &&d(x,y,u)=\exp(y)-u,\\ &&y_2(x)=0,\quad \Gamma=\Gamma_2,\qua... ...\pi x_1)\sin(2\pi x_2)\\ &&\psi(x)=.11,\quad u_1=-5,\quad u_2=5 \end{eqnarray*}$$

Table 12: Minimal eigenvalue for (ED-3)

$1/h$	$\gamma_h$	$\gamma_h/h^2$
40	6.250e-7	1.0e-3
50	4.000e-7	1.0e-3
60	2.777e-7	1.0e-3

(ED-4)

$$\begin{eqnarray*} &&d(x,y,u)=\exp(y)-u,\quad b(x,y)=-y,\quad \Gamma=\Gamma_1,\qu... ...i x_1)\sin(2\pi x_2),\quad \psi(x)=.371,\quad u_1=-8,\quad u_2=9 \end{eqnarray*}$$

Table 13: Minimal eigenvalue for (ED-4)

$1/h$	$\gamma_h$	$\gamma_h/h^2$
40	6.250e-7	1.0e-3
50	4.000e-7	1.0e-3
60	2.777e-7	1.0e-3

(ED-5) as (ED-1) except

$$\begin{eqnarray*} &&b(x,y)=-\beta y,\quad \beta=1\mbox{ on }x_1=0,\quad x_2=0,\q... ...ad\mbox{otherwise}\\ &&\psi(x)=6.09,\quad u_1=1.4,\quad u_2=1.6 \end{eqnarray*}$$

Table 14: Minimal eigenvalue for (ED-5)

$1/h$	$\gamma_h$	$\gamma_h/h^2$
40	1.136e-3	1.82
50	7.447e-4	1.86
60	5.399e-4	1.94