Discretization of the distributed control problem

The optimization variable $\,z\,$ in (1) is now taken as the vector

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

Due to the Dirichlet conditions (28), the variables $\,y_{ij} \,$ for $\,(i,j) \in I(\Gamma_2)\,$ can be eliminated from the optimization process.
The application of the five-point-star to the elliptic equation $\, - \Delta y(x) + d(x,y(x),u(x)) = 0 \,$ in (26) yields the following equations for all $\,(i,j) \in I(\Omega)$:

$$\hspace*{-8mm} G^h_{ij}(z):= 4y_{ij} - y_{i+1,j} - y_{i-1,j}... ...i,j+1} - y_{i,j-1} + h^2\,d(x_{ij},y_{ij},u_{ij}) = 0 \,. \;$$ (18)

The Dirichlet condition (28) is incorporated by fixing the values $\,y_{ij} \,$ on $\,\Gamma _2\,$:

$$\hspace*{15mm} y_{ij} = y_2(x_{ij}) \qquad \forall \;\; (i,j) \in I(\Gamma_2)\,.$$ (19)

Observing the approximation (5) of the outward normal derivative, the discrete form of the Neumann boundary condition in (27) leads to the equality constraints

$$B^h(x_{ij},y_{ij}):= y^{\nu}_{ij} - h\,b(x_{ij},y_{ij}) = 0\quad \mbox{for} \quad (i,j) \in I(\Gamma_1)\, . \quad$$ (20)

The control and state inequality constraints (29) and (30) yield the inequality constraints

$$C(x_{ij},y_{ij},u_{ij}) \leq 0 \,, \quad \forall \quad (i,j) \in I(\Omega)\,.$$ (21)

$$S(x_{ij},y_{ij}) \leq 0 \,, \hspace*{12mm} \forall \quad (i,j) \in I(\Omega)\,,$$ (22)

Note again that these inequality constraints do not depend on the meshsize $\,h\,$. The discretized form of the cost function (25) is

$$F^h(z):= h^2 \sum_{(i,j)\in I(\Omega)} f(x_{ij},y_{ij},u_{ij}) \,+\, h \sum_{(i,j) \in I(\Gamma_1)} g(x_{ij},y_{ij}) \,.$$ (23)

Hence, for distributed control problems the NLP-problem (1) is given by the relations (18)-(23). The corresponding Lagrangian function is

$$L(z,q,\lambda,\mu):= h^2 \sum_{(i,j)\in I(\Omega)} f(x_{ij},y_{ij},u_{ij}) \,+\, h \sum_{(i,j) \in I(\Gamma_1)} g(x_{ij},y_{ij})$$ (24)

$$\hspace*{-25mm} +\,\sum_{(i,j)\in I(\Omega)} [\,q_{ij} G^h_{ij}(z) \, +\,\lambda_{ij} C(x_{ij},y_{ij},u_{ij}) + \,\mu_{ij} S(x_{ij},y_{ij})\,]$$

$$\hspace*{-25mm} +\, \sum_{(i,j)\in I(\Gamma_1)} q_{ij} B^h(z) \,,$$

where the Lagrange multipliers $\,q=(q_{ij})_{(i,j) \in I(\Omega\cup\Gamma_1)}\,$, $\,\lambda=(\lambda_{ij})_{(i,j) \in I(\Omega)}\,$ and $\,\mu=(\mu_{ij})_{(i,j) \in I(\Omega})\,$ are associated with the equality constraints (18) and (20), resp. the inequality constraints (21) and (22). The multipliers $\,\lambda\,$ and $\,\mu\,$ satisfy complementarity conditions corresponding to (37):

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

The discussion of the necessary conditions of optimality

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

is similar to that for boundary control problems. For state variables $\,y_{ij} \,$ with indices $\, (i,j) \in I(\Omega)\,$ we obtain the relations

$$\hspace*{-5mm} 0 = L_{y_{ij}} = 4q_{ij} - q_{i+1,j} - q_{i-1,j} - q_{i,j+1} - q_{i,j-1} + h^2 q_{ij} \,d_y(x_{ij},y_{ij},u_{ij})\,+$$

$$\hspace*{-5mm} +\,h^2 \,f_y(x_{ij},y_{ij},u_{ij}) \,+\, \lambda_{ij}\,C_y(x_{ij},y_{ij},u_{ij}) +\mu_{ij}\,S_y(x_{ij},y_{ij}) \,.$$ (25)

Here as in (12), the undefined multipliers are set to

$$q_{ij} = 0 \qquad \forall \; (i,j) \in \Gamma_2 \,,$$ (26)

in accordance with the Dirichlet condition (35). We deduce from equations (25) that the Lagrange multipliers $\,q=(q_{ij})\,$ satisfy the five-point-star difference equations for the adjoint equation $\, -\Delta \bar{q} + \bar{q}\,d_y + f_y + \bar{\lambda} C_y + S_y\,\bar{\mu} = 0\;$ in (33) if we approximate the measure $\,\bar{\mu}\,$ and the multiplier function $\,\bar{\lambda}\,$ in $\,\Omega \,$ by

$$h^2\,\bar{\lambda}(x_{ij})\, \sim\ \,\lambda_{ij}\,, \quad \int_{sq(h^2)} \, d\bar{\mu} \; \sim \; \mu_{ij} \,,$$ (27)

where $\,sq(h^2)\,$ denotes a square centered at $\,x_{ij}\,$ with area $h^2$. Recall again the decomposition (38) of the measure $\,\bar{\mu} = \bar{\nu} \cdot dx\,+\,\bar{\nu}_s \cdot \bar{\mu}_s\,$. If the singular part of the measure vanishes, then (27) yields an approximation for the density $\bar{\nu}$,

$$\hspace*{20mm} \bar{\nu}(x_{ij})\, \sim\ \mu_{ij}/h^2 \,.$$ (28)

while for a delta distribution $\, \bar{\mu} = \bar{\nu}_s \cdot \delta(x-x_{ij})\,$ we deduce from (27) the approximation

$$\hspace*{20mm} \bar{\nu}_s \, \sim\ \mu_{ij} \,.$$ (29)

For indices $\,(i,j) \in I(\Gamma_1)\,$ on the boundary $\,\Gamma_1\,$ we obtain, e.g., for $\,j=0,\,i \in \{1,...,N\}\,$,

$$0= L_{y_{i0}} = -q_{i1} + q_{i0} -q_{i0}\,h\,b_y(x_{i0},y_{i0}) +h\,g_y(x_{i0},y_{i0}) \,,$$

which constitutes the discrete version of the Neumann boundary condition (34).

Finally, necessary conditions with respect to the control variables $\,u_{ij}\,$ for $\, (i,j) \in I(\Omega)\,$ are determined by

$$0= L_{u_{ij}} = h^2 f_u(x_{ij},y_{ij},u_{ij}) + q_{ij}\,h^2... ...,y_{ij},u_{ij}) + \lambda_{ij}\,C_u(x_{ij},y_{ij},u_{ij}) \,.$$

>From this equation we can recover the discrete version of the control law (12), if we use the identification

$$\bar{\lambda}(x_{ij}) \,\sim\, \lambda_{ij}/h^2 \, \qquad \forall \;\; (i,j) \in \Omega \,.$$ (30)