On the Optimal Control of Impulsive Hybrid Systems On Riemannian   Manifolds
  This paper provides a geometrical derivation of the Hybrid Minimum Principle (HMP) for autonomous impulsive hybrid systems on Riemannian manifolds, i.e. systems where the manifold valued component of the hybrid state trajectory may have a jump discontinuity when the discrete component changes value. The analysis is expressed in terms of extremal trajectories on the cotangent bundle of the manifold state space. In the case of autonomous hybrid systems, switching manifolds are defined as smooth embedded submanifolds of the state manifold and the jump function is defined as a smooth map on the switching manifold. The HMP results are obtained in the case of time invariant switching manifolds and state jumps on Riemannian manifolds. 
Farzin Taringoo, Peter E. Caines
math.OC