  We present high order explicit geometric integrators to solve
linear-quadratic optimal control problems and $N$-player differential games.
These problems are described by a system coupled non-linear differential
equations with boundary conditions. We propose first to integrate backward in
time the non-autonomous matrix Riccati differential equations and next to
integrate forward in time the coupled system of equations for the Riccati and
the state vector. This can be achieved by using appropriate splitting methods,
which we show they preserve most qualitative properties of the exact solution.
Since the coupled system of equations is usually explicitly time dependent, a
preliminary analysis has to be considered. We consider the time as two new
coordinates, and this allows us to integrate the whole system forward in time
using splitting methods while preserving the most relevant qualitative
structure of the exact solution. If the system is a perturbation of an exactly
solvable problem, the performance of the splitting methods considerably
improves. Some numerical examples are also considered which show the
performance of the proposed methods.
