Bifurcation preserving discretisations of optimal control problems

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