The basis of the direct collocation approach
is a finite dimensional approximation of control and
state variables, i. e., a discretization.
Here, we choose a continuous, piecewise linear
control approximation and a continuously differentiable,
piecewise cubic state approximation,
cf. Hargraves, Paris  and
The differential equations, the state and control
constraints are only pointwise fulfilled in this approach.
The discretization results in a nonlinear
optimization problem subject to nonlinear
Convergence properties of the method
and details of an
efficient implementation are discussed
in , .
Here, we used the code DIRCOL 
where the resulting nonlinear programming problems
are solved by the Sequential Quadratic Programming
method NPSOL due to Gill, Murray, Saunders, and Wright .
The direct collocation
method has a large domain of convergence and
is easy to handle as the user has not to
be concerned with adjoint variables or necessary conditions