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## Necessary first order optimality conditions of the discretized problem

For the sake of simplicity, we now assume that n = 1 and l = 1. In this section, we will use the notations

and

The Lagrangian of the nonlinear program of the discretized problem from Sec. 2 can then be written as

with , and . A solution of the nonlinear program fulfills the necessary first order optimality conditions of Karush, Kuhn, and Tucker, cf., e. g., [9]. Among others, these are

As the ``fineness'' of the grid, we define

In detail, we find for ,

Using the basic relations (23) -- (31) of [18] and the notation from (21), (22), we obtain after some calculations and by using the chain rule of differentiation

Letting and keeping fixed, we have

and finally

This equation is equivalent to the condition (19).
On the other hand, for ,

Using again the basic relations (23) -- (31) of [18] and the notation from (21), (22), we obtain after some calculations and by using the chain rule of differentiation

For convenience, we now suppose an equidistant grid, i.e.\

Now letting and keeping fixed, we have (cf. [18])

This equation is equivalent to the adjoint differential equation (18).
Similar results hold for a non-equidistant grid under additional conditions and for n > 1. They can also be extended to more general problems.

Oskar von Stryk
Fri Apr 5 21:38:03 MET DST 1996