Equivalence of Finite- and Fixed-time Stability to Asymptotic Stability
This work addresses theoretical connections between convergence rates in control theory, but it appears incremental as it builds on existing LaSalle-like principles without introducing a new paradigm.
The paper tackles the problem of finite- and fixed-time convergence in dynamical systems by developing non-smooth Lyapunov-like results, showing that globally asymptotically stable systems can be modified to achieve fixed-time stability through scaling.
In this paper, we present new results on finite- and fixed-time convergence for dynamical systems using LaSalle-like invariance principles. In particular, we provide first and second-order non-smooth Lyapunov-like results for finite- and fixed-time convergence, thereby relaxing the requirement of existence a differentiable, positive definite Lyapunov function. Based on these findings, we show that a dynamical system whose equilibrium point is globally asymptotically stable can be modified through scaling so that the resulting dynamical system has a fixed-time stable equilibrium point. The results in this paper expand our understanding of various convergence rates and strengthen the hypothesis that all the convergence rates are interconnected through a suitable transformation.