Bifurcation preserving discretisations of optimal control problems

TL;DR

This paper demonstrates that structure-preserving discretisation methods, especially symplectic integrators, maintain bifurcation structures in optimal control problems, unlike non-symplectic schemes which can break these bifurcations.

Contribution

It shows that generic bifurcations in optimal control problems are preserved under discretisation when using symplectic methods, providing theoretical and practical insights.

Findings

Symplectic discretisation preserves bifurcation structures.

Non-symplectic schemes can break bifurcations.

Illustration on ellipsoid cut locus demonstrates the phenomenon.

Abstract

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.