Published in Kang, W., Xiao, M., & Tall, I. A. (2003). Controllability and local accessibility - A normal form approach. IEEE Transactions on Automatic Control, 48(10), 1724-1736. doi: 10.1109/TAC.2003.817924. ©2003 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.


Given a system with an uncontrollable linearization at the origin, we study the controllability of the system at equilibria around the origin. If the uncontrollable mode is nonzero, we prove that the system always has other equilibria around the origin. We also prove that these equilibria are linearly controllable provided a coefficient in the normal form is nonzero. Thus, the system is qualitatively changed from being linearly uncontrollable to linearly controllable when the equilibrium point is moved from the origin to a different one. This is called a bifurcation of controllability. As an application of the bifurcation, systems with a positive uncontrollable mode can be stabilized at a nearby equilibrium point. In the last part of this paper, simple sufficient conditions are proved for local accessibility of systems with an uncontrollable mode. Necessary conditions of controllability and local accessibility are also proved for systems with a convergent normal form.