Variational Integrators for a new Lagrangian Approach to Control Affine Systems with a Quadratic Lagrange Term

Abstract In this work, we analyse the discretisation of a recently proposed new Lagrangian approach to optimal control problems of affine-controlled second-order differential equations with cost functions quadratic in the controls. We propose exact discrete and semi-discrete versions of the problem, providing new tools to develop numerical methods. Discrete necessary conditions for optimality are derived, and their equivalence with the continuous version is proven. As an example, a family of low-order integration schemes is devised to find approximate optimality conditions, which are used to solve both a low-thrust orbital transfer and satellite alignment problem, including a convergence and performance study. Non-trivial equivalent standard direct methods are constructed. Noether’s theorem and symplecticity for the new Lagrangian approach are investigated in the exact and approximate cases.