Example 3

In this example we choose another nonlinear partial differential equation and homogeneous Dirichlet boundary conditions:

$$\begin{eqnarray*} \hspace*{-11mm} \begin{array}{lll} \mbox{on} \;\;\Omega \,: & ... ...) = 0, &\; u_d(x) \equiv 0 \,, \; \alpha = 0.001 . \end{array}\end{eqnarray*}$$

This problem has been considered in [16], but without state constraints.

Table 3: Information on solution of Example 3

N+1	it	CPU	Acc	$F(\bar{y})$
50	29	131	8	.110242
100	32	2257	8	.110263
200	31	42644	8	.110269

Figure 5: Example 3, $\,\alpha =0.001$ : Optimal control.

Figure 6: Example 3, $\,\alpha =0.001$ : Optimal state and adjoint variable

The adjoint equations (2.10), (2.12) yield

$$\hspace*{-10mm} \begin{array}{rlll} -\Delta \bar{q}(x) - \ba... ...(x) & = 0 & \quad \mbox{on} & \;\; \Gamma \,. \quad \end{array}$$ (4.4)

The minimum condition (2.21) leads again to the projection

$$\bar{u}(x) = P_{\,[u_1,u_2]}\,(\bar{q}(x)/\alpha)$$ (4.5)

with $\,[u_1,u_2] = [-5,5]\,$. The optimal control is shown in Figure 5, while Figure 6 displays the optimal state and associated adjoint variable. The adjoint variable allows to verify the control rule (4.5). Note that condition (2.7) does not hold for this example in view of $\,d_y(x,\bar{y}(x)) = -\exp(\bar{y}(x)) < 0 \,$.
The state is active in the points $\,x^1=(.25,.25), \,x^2=(.75,.75)\,$. In view of the decomposition (2.15), the measure $\,\bar{\mu}\,$ is given here by the singular measure $\,\,\bar{\mu} = \bar{\nu}_s^1 \cdot \delta(x-x^1) + \bar{\nu}_s^2 \cdot \delta(x-x^2)\,$, where $\,\bar{\nu}_s^1,\,\bar{\nu}_s^2\,$ are identified with the multipliers $\,\mu_{ij}\,$ in view of (3.15). Let us check now the accuracy with which the computed adjoint equation is satisfied. At the point $\,x_{ij}=x^1=(.25,.25)\,$ we obtain $\,q_{ij} = .0085852, \,q_{i+1,j}= .0092638, \,q_{i-1,j} = .0091208, \, q_{i,j+1... ...j-1} = .0091208, \, y_{ij} = .11, \, y_d(x_{ij}) = 1, \, \mu_{ij} = .0025181 \,$. Then the discretized adjoint equation (3.11) yields $\, 4 q_{ij} - q_{ij} - q_{i+1,j} - q_{i-1,j} - q_{i,j+1} - h^2 q_{ij} \exp(y_{ij}) + h^2 (y_{ij} - y_d(x_{ij})) + \mu_{ij} \,= -0.0000003 \,$ for $\,h=1/100\,$.