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Dirichlet boundary conditions

The Dirichlet boundary conditions in (3.2) are incorporated by the relations
y_{ij} = b(x_{ij},u_{ij}) \quad \mbox{for all} \quad (i,j) \in I(\Gamma)\,.
\end{displaymath} (4.15)

Hence it suffices to work with a reduced number of optimization variables in (4.1), namely we can take

z:=(\,(y_{ij})_{\,(i,j) \in I(\Omega)}, \,(u_{ij})_{\,(i,j) \in I(\Gamma)}\,)
\in I\!\! R^{\,N^2+4N} \,.

The equality constraints agree with those from (4.3),
G^h_{ij}(z):= 4y_{ij} - y_{i+1,j} - y_{i-1,j} - y_{i,j+1} - y_{i,j-1}
+ h^2\,d(x_{ij},y_{ij}) = 0 \,
\end{displaymath} (4.16)

for all indices $\,(i,j) \in I(\Omega)$ where the values $\,y_{ij} \,$ on the boundary are substituted from the relations (4.15). The control and state inequality constraints (3.3) resp. (3.4) give rise to the inequality constraints:
    $\displaystyle S(x_{ij},y_{ij}) \leq 0 \,, \quad (i,j) \in I(\Omega)\,,$ (4.17)
    $\displaystyle C(x_{ij},u_{ij}) \leq 0 \,, \quad (i,j) \in I(\Gamma)\,.$ (4.18)

Finally, the cost function is derived from (3.1) as
F^h(z):= h^2 \sum_{(i,j)\in I(\Omega)} f(x_{ij},y_{ij}) \,+\,
h \sum_{(i,j) \in I(\Gamma)} g(x_{ij},u_{ij}) \,.
\end{displaymath} (4.19)

By means of (4.16)-(4.19) we have obtained an NLP-problem of the form (4.1).

Let us evaluate the first order optimality conditions of Kuhn-Tucker type for problem (4.16)-(4.19). The Lagrangian function becomes

$\displaystyle \hspace*{-10mm}
L(z,q,\mu,\lambda):=$   $\displaystyle \hspace*{-4mm}
h^2 \sum_{(i,j)\in I(\Omega)} f(x_{ij},y_{ij}) \,+\,
h \sum_{(i,j) \in I(\Gamma)} g(x_{ij},u_{ij})$ (4.20)
    $\displaystyle \hspace*{-6mm}
+\,\sum_{(i,j)\in I(\Omega)} [\,q_{ij} G^h_{ij}(z)...
+\, \sum_{(i,j)\in I(\Gamma)} \lambda_{ij} C(x_{ij},u_{ij}) \,,$  

where the Lagrange multipliers $\,q=(q_{ij})_{(i,j) \in I(\Omega)}\,$, $\,\mu=(\mu_{ij})_{(i,j) \in I(\Omega)}\,$ and
$\,\lambda=(\lambda_{ij})_{(i,j) \in I(\Gamma)}\,$ which are associated with the equality constraints (4.16), the inequality constraints (4.17), and the inequality constraints (4.18). The multipliers $\,\lambda\,$ and $\,\mu\,$ satisfy the complementarity conditions corresponding to (3.9):

\lambda_{ij} \geq 0 & \mbox{and} & \; \lam...
...}) = 0
& \mbox{for all} \quad (i,j) \in I(\Omega) \,.

In the next step we discuss the necessary conditions of optimality

\,0 = L_z = (\,(L_{y_{ij}})_{(i,j)\in I(\Omega)}\,,
(L_{u_{ij}})_{(i,j)\in I(\Gamma)}\,)

for different combinations of indices $\,(i,j)\,$. For indices $\, 2 \leq i,j \leq N-1 \,$ we obtain the relations

0 = L_{y_{ij}} = && \hspace*{-4mm}
4q_{ij} - q_{i+1,j} - q_{i-...
...+ h^2 \,f_y(x_{ij},y_{ij})\,+\,

Hence, for this set of indices we see that the Lagrange multipliers $\,q=(q_{ij})\,$ satisfy the five-point-star difference equations for the adjoint equation $\, -\Delta \bar{q} + \bar{q}\,d_y + f_y + S_y\,\bar{\mu}= 0 \,$ in (3.6) if we use the same identification for the Borel measure $\,\bar{\mu}\,$ as in (4.10),
\int_{s(h^2)} d\bar{\mu} \, \sim \, \mu_{ij} \,.
\end{displaymath} (4.21)

from which we also get the special cases (4.11) and (4.12). The situation is different for indices with either $\,i=1, \,j=1, \,i=N\,$ or $\,j=N$. E.g., for indices $\,j=1, \,i=2,...,N-1,\,$ we get

0= L_{y_{i1}} = && \hspace*{-4mm}
4q_{i1} - q_{i+1,1} - q_{i...
...+ h^2 \,f_y(x_{i1},y_{i1})\,+\,

Note that this equation does not involve the variable $\,q_{i0}\,$ on the boundary. Defining $\,q_{i0}=0\,$ we arrive again at the discretized form of the adjoint equation (3.6) using approximations as in (4.10)-(4.12). Similar relations hold for index combinations $\,i=1, \,j=2,..,N-1, \,$ or $\,i=N, \,j=2,..,N-1,\,$ or $\,j=N, \,i=2,..,N-1\,$. For the remaining indices we find, e.g. for $\,i=j=1\,$:

0= L_{y_{11}} = 4q_{11} - q_{21} - q_{12}
+ h^2 q_{11} \,d...
... h^2 \,f_y(x_{11},y_{11})\,+\,
\mu_{11}S_y(x_{11},y_{11}) \,.

These relations do not involve the boundary variables $\,q_{10}\,$ and $\,q_{01}\,$. Again, defining $\,q_{10}=0\,$ and $ \,q_{01}=0\,$ we recover the adjoint equations. In summary, by requiring the Dirichlet boundary condition for the adjoint variables,

q_{ij} = 0 \quad \mbox{for} \quad (i,j) \in I(\Gamma),

we see that the adjoint equation holds for all indices $\,(i,j) \in I(\Omega)\,.$

The necessary conditions with respect to the control variables $\,u_i\,$ are determined for the indices $j=0,\,i=1,...,N,$ by

0= L_{u_{i0}}
&& \hspace*{-5mm}
= h g_u(x_{i0},u_{i0}) - q_{i1...
...x_{i0},u_{i0}) +

Observing $\,q_{i0}=0\,$ and the approximation of the normal derivative in (4.4), the minimum condition (3.8) holds with the identifications
\partial_{\nu}\bar{q}(x_{i0}) \,\sim \, - q_{i1}/h \,, \quad
\bar{\lambda}(x_{i0}) \,\sim \, \lambda_{i0} / h
\end{displaymath} (4.22)

Similar identifications hold for the other indices $\,(i,j) \in I(\Gamma)\,$.

next up previous
Next: Optimization codes and modeling Up: Discretization and optimization techniques Previous: Neumann boundary conditions
Hans D. Mittelmann