This section briefly recalls the discretization scheme as described in more detail
Some of the basic ideas of this discretization scheme have been formerly outlined by
Kraft  and
Hargraves and Paris .
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
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.