Example 1

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

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

These equations are related to a simplified Ginzburg-Landau model for super-con- ductivity in the absence of internal magnetic fields with $\,y\,$ the wave function; cf. Ito, Kunisch [19] and Kunisch, Volkwein [20].

The adjoint equations (2.10), (2.12) apply:

$$\hspace*{-10mm} \begin{array}{rlll} -\Delta \bar{q}(x) + \ba... ...(x) & = 0 & \quad \mbox{on} & \;\; \Gamma \,. \quad \end{array}$$ (4.1)

Since $\,u_d = 0\,$, the minimum condition (2.21) yields the control law

$$\bar{u}(x) = P_{\,[u_1,u_2]}\,(\bar{q}(x)/\alpha) \,,$$ (4.2)

where $\,P_{\,[u_1,u_2]}\,$ denotes the projection operator onto the interval $\,[u_1,u_2] = [1.5,4.5]\,$. The optimal control is shown in Figure 1, while Figure 2 depicts the optimal state and associated adjoint variable. The adjoint variable permits the verification of the control law (4.2). The state is active in the center $\,(0.5,0.5)\,$. Figure 2 indicates that condition (2.7) is not satisfied since $\,d_y(x,\bar{y}(x)) = 3 \bar{y}(x)^2 - 1 < 0 \,$ holds for all $x \in \Omega$. However, well-posedness of the Ginzburg-Landau model follows from results in Gunzburger et al. [15].

Figure 1: Example 1 $\,\alpha =0.001$ : Optimal control.

Figure 2: Example 1, $\,\alpha =0.001$ : Optimal state and adjoint variable

Table 1: Information on solution of Example 1

N+1	it	CPU	Acc	$F(\bar{y})$
50	26	104	8	.0577903
100	30	1897	8	.0621615
200	35	54831	8	.0644259

Figure 3: Example 2, $\,\alpha =0$ : Optimal control and switching curve $\,\bar{q}(x) = 0\,$.

Figure 4: Example 2, $\,\alpha =0$ : Optimal state and adjoint variable