Improving extensions for the global uniform asymptotic stability of continuous-time nonlinear systems

This paper considers the classical problems of global asymptotic stability (GAS) of nonlinear autonomous (time-invariant) systems and global uniform asymptotic stability (GUAS) of nonlinear non-autonomous (time-varying) systems. By imposing new conditions on the Lyapunov function and its derivative, the state convergence to the origin from any initial condition with bounded norm in the classical setting is extended to the initial conditions with unbounded norms in the present work. The proposed new notions of stability are called extended global asymptotic stability (EGAS) and extended global uniform asymptotic stability (EGUAS) in this paper. The proposed Lyapunov-based criteria yield the global asymptotic stability of the origin and fixed-time/predefined-time attractiveness of any selected neighbourhood of the origin. The result is that the dynamical behaviour of the state trajectories from the convergence time point of view is improved considerably. The relationship between the smoothness and input-to-state stability (ISS) properties is also studied from a quantitative point of view. Under some new sufficient conditions, it is shown that dynamical systems with smaller Lipschitz constants represent smaller ultimate bounds, and as a result, are more robust. The desirable dynamical behaviour and robustness property of the proposed stability notion makes it comparable with the finite/fixed/predefined/prescribed (exact) stable systems that are naturally non-Lipschitz at the origin. Numerical results illustrate the effectiveness of the proposed theoretical results.