Geometric integrators in optimal control theory

Geometric integrators are structure-peserving integrators with the goal to capture the dynamical system's behavior in a most realistic way. Using structure-preserving methods for the simulation of mechanical systems, specific properties of the underlying system are handed down to the numerical solution, for example, the energy of a conservative system shows no numerical drift or momentum maps induced by symmetries are preserved exactly. One particular class of geometric integrators is the class of variational integrators. They are derived from a discrete variational principle based on a discrete action function that approximates the continuous one. The resulting schemes are symplectic-momentum conserving and exhibit good energy behaviour. 

 

For the numerical solution of optimal control problems, direct methods are based on a discretization of the underlying differential equations which serve as equality constraints for the resulting finite dimensional nonlinear optimization problem. For the case of mechanical systems, we use variational integrators for the discretization of optimal control problems. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators optimization and discretization commute due to the symplecticity of the discretization scheme. This property guarantees that the convergence rates are preserved for the adjoint system which is also referred to as the Covector Mapping Principle.