Dirichlet boundary conditions

The Dirichlet boundary conditions in (3.2) are incorporated by the relations

$$\hspace*{15mm} y_{ij} = b(x_{ij},u_{ij}) \quad \mbox{for all} \quad (i,j) \in I(\Gamma)\,.$$ (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),

$$\hspace*{-5mm} G^h_{ij}(z):= 4y_{ij} - y_{i+1,j} - y_{i-1,j} - y_{i,j+1} - y_{i,j-1} + h^2\,d(x_{ij},y_{ij}) = 0 \,$$ (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:

$$S(x_{ij},y_{ij}) \leq 0 \,, \quad (i,j) \in I(\Omega)\,,$$ (4.17)

$$C(x_{ij},u_{ij}) \leq 0 \,, \quad (i,j) \in I(\Gamma)\,.$$ (4.18)

Finally, the cost function is derived from (3.1) as

$$F^h(z):= h^2 \sum_{(i,j)\in I(\Omega)} f(x_{ij},y_{ij}) \,+\, h \sum_{(i,j) \in I(\Gamma)} g(x_{ij},u_{ij}) \,.$$ (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

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

$$\hspace*{-6mm} +\,\sum_{(i,j)\in I(\Omega)} [\,q_{ij} G^h_{ij}(z)... ...{ij},y_{ij})\,] +\, \sum_{(i,j)\in I(\Gamma)} \lambda_{ij} C(x_{ij},u_{ij}) \,,$$

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):

$$\begin{eqnarray*} \begin{array}{llll} \lambda_{ij} \geq 0 & \mbox{and} & \; \lam... ...}) = 0 & \mbox{for all} \quad (i,j) \in I(\Omega) \,. \end{array}\end{eqnarray*}$$

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

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

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),

$$\hspace*{20mm} \int_{s(h^2)} d\bar{\mu} \, \sim \, \mu_{ij} \,.$$ (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

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

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

$$\begin{eqnarray*} 0= L_{u_{i0}} && \hspace*{-5mm} = h g_u(x_{i0},u_{i0}) - q_{i1... ...x_{i0},u_{i0}) + \frac{\lambda_{i0}}{h}\,C_u(x_{i0},u_{i0})\,] \end{eqnarray*}$$

Observing $\,q_{i0}=0\,$ and the approximation of the normal derivative in (4.4), the minimum condition (3.8) holds with the identifications

$$\partial_{\nu}\bar{q}(x_{i0}) \,\sim \, - q_{i1}/h \,, \quad \bar{\lambda}(x_{i0}) \,\sim \, \lambda_{i0} / h$$ (4.22)

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