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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:

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

Since $\,u_d = 0\,$, the minimum condition (2.21) yields the control law
\begin{displaymath} \bar{u}(x) = P_{\,[u_1,u_2]}\,(\bar{q}(x)/\alpha) \,, \end{displaymath} (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.
\begin{figure}\centerline{\hbox{ \epsfig{figure=fig4.1a.ps,height=3in,width=8cm} }}\end{figure}

Figure 2: Example 1, $\,\alpha =0.001$ : Optimal state and adjoint variable
\begin{figure}\centerline{\hbox{ \epsfig{figure=fig4.1b.ps,height=3in,width=8cm} \epsfig{figure=fig4.1c.ps,height=3in,width=8cm} }}\end{figure}


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 $\,\bar{q}(x) = 0\,$.
\begin{figure}\centerline{\hbox{ \epsfig{figure=fig4.2a.ps,height=3in,width=8cm} \epsfig{figure=fig4.2b.ps,height=3in,width=8cm} }}\end{figure}

Figure 4: Example 2, $\,\alpha =0$ : Optimal state and adjoint variable
\begin{figure}\centerline{\hbox{ \epsfig{figure=fig4.2c.ps,height=3in,width=8cm} \epsfig{figure=fig4.2d.ps,height=3in,width=8cm} }}\end{figure}

next up previous
Next: Example 2 Up: Numerical examples Previous: Numerical examples
Hans D. Mittelmann
2000-10-06