The minimization of the phase noise 
with respect to the design parameters
and subject to Eqs. (
) -- ()
is an optimal control problem with 
as (constant) control function
from a finite dimensional control space
and 
and 
as state variables
of the optimal control problem.
The optimization problem is solved numerically by
the direct collocation method DIRCOL [].
The method is based on a discretization of the state variables
by piecewise cubic spline functions
satisfying the dynamic equations at the
grid points of a time grid and at the
centers between
(cf. Hargraves, Paris []
and []).
By this approach, the optimal control problem
is transformed to a (finite dimensional)
nonlinearly constrained optimization problem
that can be solved by the Sequential Quadratic Programming
method due to Gill et al. [].
Compared with other methods for solving optimal control problems
the direct collocation method
is easy-to-use (as knowledge of optimal control theory is not
required), robust (not much information on the solution is a priori required)
and reliable (if accuracy requirements are not too high) [].