Lifting of trajectories of control systems related by smooth mappings

Abstract

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.