Discretization and optimization techniques

The discussion of discretization schemes is restricted to the standard situation where the domain is the unit square $\,\Omega=(0,1)\times (0,1)\,.$ The purpose of this section is to develop discretization techniques by which the boundary control problem (1)-(6) and the distributed control problem (25)-(30) are transformed into a nonlinear programming problem (NLP-problem) of the form

$$\mbox{Minimize} \quad F^h(z) \quad \mbox{subject to} \quad G^h(z) = 0 \,, \quad H(z) \leq 0 \,.$$ (1)

The functions $\,F^h, G^h\,$ and $\,H\,$ are sufficiently smooth and are of appropriate dimension. The upper subscript $\,h\,$ denotes the dependence on the stepsize. The optimization variable $\,z\,$ will comprise both the discretized state and control variables.

The form (1) will be achieved by solving the elliptic equation (2) resp. (26) with the standard five-point-star discretization scheme. Choose a number $\,N \in \rm I\! N_+\,$ and the stepsize $\,h:=1/(N+1)\,.$ Consider the mesh points

$$\,x_{ij}=(ih,jh)\,, \quad 0 \leq i,j \leq N+1, \,$$

and define the following sets of indices $\,(i,j)\,$ residing either in the domain $\,\Omega \,$ or on the four edges of the boundary $\,\Gamma\,$:

$$\begin{array}{l} I(\Omega):=\{\,(i,j)\, \vert \; 1 \leq i,j ... ...mega\cup\Gamma_1):= I(\Omega) \cup I(\Gamma_1) \, . \end{array}$$ (2)

Obviously, we have $\;\char93 I(\Omega)= N^2\,, \;\char93 I(\Gamma)= 4*N\,$; define further $\;M_1:=\char93 I(\Gamma_1) \,$.

We shall first discuss discretization schemes for the boundary control problem and will then only indicate the necessary modifications to obtain schemes for the distributed control problem.