Abstract. Lyapunov-like characterizations for the concepts of nonuniform in time robust global asymptotic stability and input-to-state stability for time-varying systems are established. The main result of our work enables us to derive (1) necessary and sufficient conditions for feedback stabilization for affine in the control systems and (2) sufficient conditions for the propagation of the input-to-state stability property through integrators.Key words. nonuniform in time asymptotic stability, input-to-state stability, Lyapunov functions, feedback stabilization AMS subject classifications. 93D20, 93D30, 37B55, 93D15 DOI. 10.1137/S03630129013929671. Introduction. The notion of nonuniform in time robust global asymptotic stability (RGAS) is basically motivated by the problem of feedback stabilization for a class of nonlinear systems that, although fail to be stabilized at a specific equilibrium by continuous static time-invariant feedback, a time-varying feedback controller can be constructed in such a way that the equilibrium for the resulting closed-loop timevarying system is asymptotically stable, in general being nonuniform with respect to the initial values of time. The notion of RGAS-without uniformity with respect to time-is also motivated by problems related to feedback stabilization, such as• stabilization of systems with uncertainties, • stabilization of systems at a reference trajectory. In the problems mentioned above, the analysis is reduced to studying asymptotic stability at a specific equilibrium of a time-varying system, whose dynamics are in general unbounded with respect to time. Particularly, in [40,41] it is shown that for a class of triangular systems whose dynamics contain time-varying unknown parameters, it is possible to find, by applying a backstepping design procedure, a smooth timevarying feedback controller in such a way that the equilibrium of the resulting closedloop system is RGAS, in general nonuniform with respect to initial values of time. Further progress has been obtained in [12,13,14,15,16,17] for a large class of nonlinear systems that in general fail to be uniformly asymptotically stabilized by smooth static time-invariant feedback at a specific equilibrium. It is worthwhile to note that among other things in the works [12,14], by employing the concept of nonuniform in time RGAS and its Lyapunov characterizations, we derive sufficient conditions for the solvability of the state feedback tracking control problem for a class of nonholonomic systems that includes the nonholonomic case in chained form. The corresponding results generalize those obtained in the literature for the same problem, since they are based on much weaker hypotheses. We finally mention the
