Lifting of trajectories of control systems related by smooth mappings

Given a pair of control systems x˙=f(x,u) and y˙=h(y,v) whose state spaces are differentiable manifolds M and N, respectively, we focus on situations where there exists a mapping Φ:M→N that induces some type of correspondence between the trajectories of the systems. An important instance of this situation occurs when f is a complex system (for example, one having many states), h is a subsystem of f (presumably, with fewer states and a simpler structure than f), and Φ is a mapping from the full state space M to a reduced state space N. Some recent research has concentrated on the concept of Φ-related systems, where it is required that Φ send trajectories of f onto trajectories of h. Here we approach the problem from a different direction and ask under what conditions trajectories of the subsystem h can be lifted to trajectories of the full system f. We provide computable sufficient conditions for the local and global lifting of trajectories from N to M. Connections between the lifting problem and the problem of global (f,g)-invariance are also discussed briefly.