On the optimal control of hybrid systems on Lie groups and the exponential gradient HMP algorithm

This paper provides a theory and associated algorithm for the optimization of autonomous and controlled hybrid systems on Lie groups. First, a geometrical derivation of the Hybrid Minimum Principle (HMP) for hybrid systems whose state manifolds constitute a Lie group (G, *) which is left invariant under the controlled dynamics of the system is presented. Second, a geometrical algorithm is developed by employing the notion of exponential curves on Lie groups. The convergence analysis for the proposed algorithm is based on Lasalle Theory. Simulation results are provided at the end of the paper.