Bifurcation preserving discretisations of optimal control problems

The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value for Hamiltonian systems. Variational or symplectic discretisation methods are classically known their excellent long term behaviour. As posed on intervals fixed, moderate length, it is not immediately clear whether profit from structure preservation in this context. When parameters present, solutions undergo bifurcations, instance, two merge and annihilate one another varied. We will show that generic bifurcations an OCP preserved under when the either directly discretised to a discrete (direct method) translated into problem using necessary which then solved integrator (indirect method). Moreover, certain break non-symplectic scheme used. general phenomenon illustrated example cut locus ellipsoid.