## Section: Research Program

### 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 [22], Bonnans [17].
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.