Global and semiglobal stabilizability in certain cascade nonlinear systems

This paper addresses the issue of global stabilization of an important class of nonlinear systems, namely, a cascade of a linear, controllable system with a nonlinear system which is asymptotically (even exponentially) stable. Such systems arise as the normal form of "minimum phase" nonlinear systems. Local stabilizability results have been known for some time for such systems. Global stabilizability results have also been obtained in some cases. However, it is also known, that when the linear "connection" to the nonlinear system is nonminimum phase, then global or semiglobal stabilizability may be impossible. Indeed, it has been shown that for any nonminimum phase linear subsystem, there exists a nonlinear asymptotically stable subsystem, for which the cascade cannot be globally stabilized. We expand on the understanding of this area by establishing, for a broader class of systems, conditions under which global or semiglobal stabilization is impossible for linear and nonlinear feedbacks.