Fast Linear Algebra for Multiarc Trajectory Optimization

This paper presents some methods for solving in a fast and reliable way the linear systems arising when solving an optimal control problem by a Runge-Kutta discretization scheme, combined with an interior-point algorithm. Our analysis holds for a multiarc problem, i.e., when several arcs, each of them associated with a dynamics and integral cost, are linked by junction points, called nodes; with the latter are associated junction conditions and a cost function.

Our main result is that a sparse QR band factorization combined with a specific elimination procedure for arcs and nodes allows to factorize the Jacobian of the discrete optimality system in a small number of operations. Combined with an “optimal” refinement procedure, this gives an efficient method that we illustrate on Goddard’s problem.