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

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

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 $\,(i,j) \in I(\Omega)$,

$$\hspace*{-8mm} G^h_{ij}(z):= 4y_{ij} - y_{i+1,j} - y_{i-1,j} - y_{i,j+1} - y_{i,j-1} + h^2\,d(x_{ij},y_{ij}) = 0 \,. \;$$ (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),

$$\hspace*{15mm} y_{ij} = a(x_{ij},u_{ij}) \qquad \mbox{for all} \quad (i,j) \in I(\Gamma_2)\,.$$ (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

$$y^{\,\nu}_{ij} := \; \left \{ \begin{array}{llll} \ y_{i0} ... ... \mbox{for} & \; j=N+1, & i=1,...,N \end{array}\right \} \, .$$ (5)

Then the discrete form of the Neumann boundary condition (3) leads to the equality constraints

$$B^h(z):= y^{\nu}_{ij} - h\,b(x_{ij},y_{ij},u_{ij}) = 0 \qquad \mbox{for} \quad (i,j) \in I(\Gamma_1)\, . \quad$$ (6)

The control and state inequality constraints (5) and (6) yield the inequality constraints

$$C(x_{ij},u_{ij}) \leq 0 \qquad \mbox{for} \quad (i,j) \in I(\Gamma)\,.$$ (7)

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

$$\hspace*{-8mm} \begin{array}{ll} F^h(z):= & h^2 \,\sum_{(i,j... ...\sum_{(i,j) \in I(\Gamma_2)}\, h(x_{ij},u_{ij}) \,. \end{array}$$ (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:

$$\hspace*{-5mm} L(z,q,\lambda,\mu): = h^2 \sum_{(i,j)\in I(\Omega)} f(x_{ij},y_{ij}) \,+\, h \sum_{(i,j) \in I(\Gamma_1)} g(x_{ij},y_{ij},u_{ij}) \quad$$ (10)

$$\hspace*{-10mm} + \, h \sum_{(i,j) \in I(\Gamma_2)} k(x_{ij},y_{i... ..._{(i,j)\in I(\Omega)} [\,q_{ij} G^h_{ij}(z) \, + \,\mu_{ij} S(x_{ij},y_{ij})\,]$$

$$\hspace*{-10mm} +\, \sum_{(i,j)\in I(\Gamma_1)} q_{ij} B^h(z) + \,\sum_{(i,j)\in I(\Gamma)} \lambda_{ij}C(x_{ij},u_{ij}) \,.$$

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

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

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

$$0 = L_{y_{ij}} = 4q_{ij} - q_{i+1,j} - q_{i-1,j} - q_{i,j+1} - q_{i,j-1} + h^2 q_{ij} \,d_y(x_{ij},y_{ij})$$

$$+\,h^2 \,f_y(x_{ij},y_{ij})\,+\, \mu_{ij}S_y(x_{ij},y_{ij}) \,.$$ (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

$$q_{ij} = 0 \qquad \mbox{for} \quad (i,j) \in I(\Gamma_2) \,.$$ (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 $\,\bar{\mu}\,$,

$$\int_{sq(h^2)} \, d\bar{\mu} \; \sim \; \mu_{ij} \,,$$ (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 $\,\bar{\nu}$,

$$\hspace*{20mm} \bar{\nu}(x_{ij})\, \sim\ \mu_{ij}/h^2 \,.$$ (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

$$\hspace*{20mm} \bar{\nu}_s \, \sim\ \mu_{ij} \,.$$ (15)

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

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

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

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

This equation yields the discrete version of the optimality condition (12) for the control, if we use the identification

$$\bar{\lambda}(x_{i0}) \,\sim\, \lambda_{i0}/h \,.$$ (16)

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

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

Observing $\,q_{i0}=0\,$ and the approximation (5) of the normal derivative, the minimum condition (13) holds with the substitutions

$$\partial_{\nu}\bar{q}(x_{i0}) \,\sim \, - q_{i1}/h \,, \quad \bar{\lambda}(x_{i0}) \,\sim \, \lambda_{i0} / h \, .$$ (17)