Simultaneous stabilization of linear and nonlinear systems by means of nonlinear state feedback

The simultaneous stabilization (resp., asymptotic stabilization) of a countable family of control systems consists of finding a control which stabilizes (resp., asymptotically stabilizes) all the systems in the family. In this paper, we introduce a new method which enables us to show that, given any countable family of stabilizable nonlinear systems, there exists a continuous state feedback law which simultaneously stabilizes (not asymptotically) the family. Then, by enriching this method, we prove that any finite family of stabilizable linear time invariant (LTI) systems can be simultaneously exponentially stabilized by means of nonlinear time-varying state feedback. We also derive sufficient conditions for the simultaneous asymptotic stabilizability of countably infinite families of LTI systems. Finally, sufficient conditions for the simultaneous asymptotic stabilizability of finite families of nonlinear systems are provided and used for the simultaneous asymptotic stabilization of certain pairs of nonlinear homogeneous systems.