Three flavors of fast convergence in dynamical systems: finite-time stability (the system reaches equilibrium in bounded time), fixed-time stability (it reaches equilibrium in time independent of initial conditions), and asymptotic stability (it converges but takes infinite time). These seem like different properties — different speeds, different guarantees, different mathematical conditions. Garg (arXiv: 2603.22802) shows they're the same thing.
The key result: any globally asymptotically stable system can be rescaled — its dynamics multiplied by a time-dependent factor — so that the resulting system has a fixed-time stable equilibrium. The convergence rate is not an intrinsic property of the dynamics. It's a property of the time coordinate.
The proof uses non-smooth Lyapunov functions and LaSalle-type invariance principles. First-order and second-order conditions for finite/fixed-time convergence are derived that relax the usual smoothness and positive-definiteness requirements on Lyapunov functions. Then the equivalence is established constructively: given an asymptotically stable system, here's the scaling that makes it fixed-time stable.
The through-claim: convergence speed is a choice, not a fact. The distinction between “gets there eventually” and “gets there in bounded time” is not a property of the dynamical system — it's a property of how you measure time. Any system that converges at all can be made to converge in any time you choose, by warping the clock. The three stability concepts are the same concept viewed through different temporal lenses. What looks like a classification of dynamical behaviors turns out to be a classification of time parameterizations.
Garg, 2603.22802. Dynamical systems / stability theory / Lyapunov methods / time scaling / fixed-time convergence.