Smooth Stabilization Implies Coprime Factorization

This paper shows that coprime right factorizations exist for the input to state mapping of a continuous time nonlinear system provided that the smooth feedback stabilization problem be solvable for this system. In particular, it follows that feedback linearizable systems admit such factorizations. In order to establish the result a Lyapunov-theoretic definition is proposed for “bounded input bounded output” stability. The main technical fact proved relates the notion of stabilizability studied in the state space nonlinear control literature to a notion of stability under bounded control perturbations analogous to those studied in operator theoretic approaches to systems; it states that smooth stabilization implies smooth input-to-state stabilization.