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# Discretization

This section briefly recalls the discretization scheme as described in more detail in [18]. Some of the basic ideas of this discretization scheme have been formerly outlined by Kraft [14] and Hargraves and Paris [11].
A discretization of the time interval

is chosen. The parameters of the nonlinear program are the values of control and state variables at the grid points and the final time

The controls are chosen as piecewise linear interpolating functions between and for

The states are chosen as continuously differentiable functions and piecewise defined as cubic polynomials between and with at ,

The approximating functions of the states have to satisfy the differential equations (2) at the grid points , and at the centers , , of the discretization intervals. This scheme is also known as cubic collocation at Lobatto points. The chosen approximation (8) -- (12) of already fulfills these constraints at . Therefore, the only remaining constraints in the nonlinear programming problem are

• the collocation constraints at

• the inequality constraints at the grid points

• and the initial and end point constraints at and

In the following, the index ``app'' for approximation will be suppressed.
By this scheme the number of four free parameters for each cubic polynomial is reduced to two and the number of three collocation constraints per subinterval is reduced to one. Compared with other collocation schemes we have

a reduced number of constraints to be fulfilled and a reduced number of free parameters to be determined by the numerical procedure. This results in a better performance of an implementation of this method in terms of convergence, reliability, and efficiency compared with other schemes.

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