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Discretization of the boundary control problem

Let $\,y_{ij} \,$ denote approximations of the state variables $\,y(x_{ij})\,$ for $\,(i,j) \in I(\bar{\Omega})\,$ and let $\,u_{ij}\,$ be approximations for the control variables $\,u(x_{ij})\,$ for $\,(i,j) \in I(\Gamma)\,$. We specify the functions $\,F^h, G^h, H\,$ for the optimization problem (1) corresponding to problem (1)-(6) as follows. The optimization variable $\,z\,$ in (1) is taken as the vector

z:=(\,(y_{ij})_{\,(i,j) \in I(\Omega\cup\Gamma_1)}\,,
...{\,(i,j) \in I(\Gamma)}\,) \,\in \rm I\! R^{\,N^2+M_1+4*N} \,.

Note that we do not consider the variables $\,y_{ij} \,, \,(i,j) \in I(\Gamma_2)\,,$ explicitly as optimization variables since they are prescribed by the Dirichlet condition (4).

Equality constraints are obtained by applying the five-point-star to the elliptic equation $\, - \Delta y(x) + d(x,y(x)) = 0 \,$ in (2) in all points $\,x_{ij}\,$ with $\,(i,j) \in I(\Omega)$,

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} (3)

In these equations may appear the undefined variables $\,y_{ij} \,$ for $ \,(i,j)\in\Gamma_2\,$. These variables have to be substituted by the Dirichlet conditions (4),
y_{ij} = a(x_{ij},u_{ij}) \qquad \mbox{for all} \quad (i,j) \in I(\Gamma_2)\,.
\end{displaymath} (4)

The derivative $\,\partial_{\nu} y(x_{ij}) \,$ in the direction of the outward normal is approximated by the expression $\, y^{\,\nu}_{ij}/h\,$ where
y^{\,\nu}_{ij} := \;
\left \{
\ y_{i0} ...
... \mbox{for} & \; j=N+1, & i=1,...,N
\end{array}\right \} \, .
\end{displaymath} (5)

Then the discrete form of the Neumann boundary condition (3) leads to the equality constraints
y^{\nu}_{ij} - h\,b(x_{ij},y_{ij},u_{ij}) = 0 \qquad \mbox{for} \quad
(i,j) \in I(\Gamma_1)\, . \quad
\end{displaymath} (6)

The control and state inequality constraints (5) and (6) yield the inequality constraints
    $\displaystyle C(x_{ij},u_{ij}) \leq 0 \qquad \mbox{for} \quad (i,j) \in I(\Gamma)\,.$ (7)
    $\displaystyle S(x_{ij},y_{ij}) \leq 0 \, \qquad \mbox{for} \quad
(i,j) \in I(\Omega)\,,$ (8)

Observe that the inequality constraints do not depend on the meshsize $\,h\,$. Later on, this fact will require a scaling of the Lagrange multipliers. Finally, the discretized form of the cost function (1) is
F^h(z):= & h^2 \,\sum_{(i,j...
...\sum_{(i,j) \in I(\Gamma_2)}\, h(x_{ij},u_{ij}) \,.
\end{array}\end{displaymath} (9)

Then the relations (2)-(8) define an NLP-problem of the form (1).

Associate Lagrange multipliers $\,q=(q_{ij})_{(i,j) \in I(\Omega\cup\Gamma_1)}\,$, $\,\lambda=(\lambda_{ij})_{(i,j) \in I(\Gamma)}\,$ and $\,\mu=(\mu_{ij})_{(i,j) \in I(\Omega)}\,$ with the equality constraints (3) and (6) resp. the inequality constraints (7) and (8). Then the Lagrangian function for the above NLP-problem becomes:

$\displaystyle \hspace*{-5mm}
L(z,q,\lambda,\mu):$   $\displaystyle = h^2 \sum_{(i,j)\in I(\Omega)} f(x_{ij},y_{ij}) \,+\,
h \sum_{(i,j) \in I(\Gamma_1)} g(x_{ij},y_{ij},u_{ij}) \quad$ (10)
    $\displaystyle \hspace*{-10mm}
+ \, h \sum_{(i,j) \in I(\Gamma_2)} k(x_{ij},y_{i...
..._{(i,j)\in I(\Omega)} [\,q_{ij} G^h_{ij}(z) \,
+ \,\mu_{ij} S(x_{ij},y_{ij})\,]$  
    $\displaystyle \hspace*{-10mm}
+\, \sum_{(i,j)\in I(\Gamma_1)} q_{ij} B^h(z)
+ \,\sum_{(i,j)\in I(\Gamma)} \lambda_{ij}C(x_{ij},u_{ij}) \,.$  

The multipliers $\,\lambda\,$ and $\,\mu\,$ satisfy complementarity conditions corresponding to (14):

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

Now we discuss the necessary conditions of optimality

\,0 = L_z = (\,(L_{y_{ij}})_{(i,j)\in I(\Omega\cup\Gamma_1) }\,,
(L_{u_{ij}})_{(i,j)\in I(\Gamma)}\,)

for state and control variables assuming different combinations of indices $\,(i,j)\,$. For state variables with indices $\, (i,j) \in I(\Omega)\,$ we obtain the relations
$\displaystyle 0 = L_{y_{ij}} =$   $\displaystyle 4q_{ij} - q_{i+1,j} - q_{i-1,j} - q_{i,j+1} - q_{i,j-1}
+ h^2 q_{ij} \,d_y(x_{ij},y_{ij})$  
    $\displaystyle +\,h^2 \,f_y(x_{ij},y_{ij})\,+\, \mu_{ij}S_y(x_{ij},y_{ij}) \,.$ (11)

These equations contain multipliers $\,q_{ij}\,$ for $ \,(i,j)\in\Gamma_2\,$ that do not appear in the Lagrangian (10). To make equations and definitions consistent, we put
q_{ij} = 0 \qquad \mbox{for} \quad (i,j) \in I(\Gamma_2) \,.
\end{displaymath} (12)

This substitution corresponds to the Dirichlet condition (11). Relations (11) then reveal 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 (9) if we use the following approximation for the Borel measure $\,\bar{\mu}\,$,
\int_{sq(h^2)} \, d\bar{\mu} \; \sim \; \mu_{ij} \,,
\end{displaymath} (13)

where $\,sq(h^2)\,$ denotes a square centered at $\,x_{ij}\,$ with area $h^2$. Recall the decomposition (15) of the measure $\,\bar{\mu} = \bar{\nu} \cdot dx\,+\,\bar{\nu}_s \cdot \bar{\mu}_s\,$. If the singular part of the measure vanishes, i.e. $ \,\bar{\nu}_s \cdot \bar{\mu}_s=0, $ then (13) yields the following appro-
ximation for the density $\,\bar{\nu}$,
\bar{\nu}(x_{ij})\, \sim\ \mu_{ij}/h^2 \,.
\end{displaymath} (14)

In case that the measure $\, \bar{\mu} = \bar{\nu}_s \cdot \delta(x-x_{ij})\,$ is a delta distribution, we obtain from (13) the relation
\bar{\nu}_s \, \sim\ \mu_{ij}
\end{displaymath} (15)

On the boundary part $\,\Gamma_1\,$ we get for indices $\,(i,j) \in I(\Gamma_1)\,$ assuming, e.g., $\,j=0,\,i \in \{1,...,N\}\,$:

0= L_{y_{i0}} =
&& -q_{i1} + q_{i0} -q_{i0}\,h\,b_y(x_{i0},y_...
...,- q_{i0}\,b_y(x_{i0},y_{i0}) +
g_y(x_{i0},y_{i0},u_{i0})\,] \,.

Recalling (5) this represents the discrete version of the Neumann boundary condition (10).

Finally, necessary conditions with respect to the control variables $\,u_{ij}\,$ for indices $\,(i,j) \in I(\Gamma)\,$ are determined by the following two relations. For $\,(i,j) \in \Gamma_1\,$ with, e.g., $j=0,\,i \in \{1,...,N\},$ we get

0= L_{u_{i0}} && = h\,g_u(x_{i0},y_{i0},u_{i0}) -
...i0},u_{i0}) +
\frac{\lambda_{i0}}{h}\,C_u(x_{i0},u_{i0})\,] \,.

This equation yields the discrete version of the optimality condition (12) for the control, if we use the identification
\bar{\lambda}(x_{i0}) \,\sim\, \lambda_{i0}/h \,.
\end{displaymath} (16)

For indices $ \,(i,j)\in\Gamma_2\,$ with, e.g., $j=0,\,i \in \{1,...,N\}\,,$ we find

0= L_{u_{i0}}
= h\,k_u(x_{i0},u_{i0}) - q_{i1}\,a_u(x_{i0}...
...x_{i0},u_{i0}) +

Observing $\,q_{i0}=0\,$ and the approximation (5) of the normal derivative, the minimum condition (13) holds with the substitutions
\partial_{\nu}\bar{q}(x_{i0}) \,\sim \, - q_{i1}/h \,, \quad
\bar{\lambda}(x_{i0}) \,\sim \, \lambda_{i0} / h \, .
\end{displaymath} (17)

next up previous
Next: Discretization of the distributed Up: Discretization and optimization techniques Previous: Discretization and optimization techniques
Hans D. Mittelmann