On Almost Lyapunov Functions for Systems with Inputs

In this work an almost Lyapunov function theorem from our recent work is generalized to systems with inputs. It is shown that if for any inputs and initial conditions, the time that solutions of the system can stay in a "bad" region where the Lyapunov function does not decrease fast enough has a sufficiently small upper bound, then the system is globally exponentially stable uniformly with respect to the inputs. In our analysis, the almost Lyapunov function is directly expressed as a function of time along arbitrary solution and the upper bound of the ratio of this function at the time the solution trajectory leaves and enters the "bad" region is found to be less than 1. Consequently all solutions are shown to converge to the origin asymptotically with some careful justification. It is also concluded that a system with inputs is exponentially input-to-state stable if its auxiliary system satisfies all the hypotheses in our main theorem. The result is then applied on an example adopted and modified from our previous work and it shows an improvement in the sense that stability can still be verified even when there is stronger perturbation to the example's stable dynamics.