Rational Stability and Characterization of Stabilizability of Nonlinear Systems

We introduce a new notion of stability for nonlinear systems called Rational Stability (RS). Rational Stability is defined for autonomous systems that are globally asymptotically stable as a property of the induced norm of a scaling operator that maps one solution of the system to another. We give an Index called Rational Stability Index (RSI) which is a positive number that quantifies the amount a uniform bound on the solution of a globally asymptotically stable nonlinear system maximally changes with respect to scaling of the initial conditions over all possible initial conditions in a given set. We present a Lyapunov-type characterization of rational stability and give an analytical tool for calculating the RSI. We provide examples of linear and nonlinear systems and after computing the RSI for them show how rational stability quantitatively reveals inherent differences in stabilizability properties of nonlinear systems.