## Section: Scientific Foundations

### Trajectory optimization

The so-called *direct methods* consist in an optimization of the trajectory,
after having discretized time, by a nonlinear programming solver that possibly
takes into account the dynamic structure.
So the two main problems are the choice of the discretization and the
nonlinear programming algorithm.
A third problem is the possibility of refinement of the discretization
once after solving on a coarser grid.

In the *full discretization approach*, general
Runge-Kutta schemes with different values of control for each inner
step are used. This allows to obtain and control high
orders of precision, see Hager [60] , Bonnans [45] .
In an interior-point algorithm context, controls can be eliminated and the
resulting system of equation is easily solved due to its band structure.
Discretization errors due to constraints are discussed in
Dontchev et al. [55] .
See also Malanowski et al. [70] .

In the *indirect* approach, the control is eliminated
thanks to Pontryagin's
maximum principle.
One has then to solve the two-points boundary value problem
(with differential variables state and costate) by a single or multiple shooting method.
The questions are here the choice of a discretization scheme for the integration of the boundary value problem, of a
(possibly globalized) Newton
type algorithm for solving the resulting finite dimensional problem
in ${I\phantom{\rule{-1.70717pt}{0ex}}R}^{n}$ ($n$ is the number of state variables), and a methodology for
finding an initial point.

For state constrained problems the formulation of the shooting function may be quite elaborated [43] , [44] . As initiated in [59] , we focus more specifically on the handling of discontinuities, with ongoing work on the geometric integration aspects (Hamiltonian conservation).