⚡ electrical engineering

Equivalence of Finite- and Fixed-time Stability to Asymptotic Stability

This paper establishes new non-smooth Lyapunov-based invariance principles for finite- and fixed-time convergence and demonstrates that any globally asymptotically stable equilibrium can be transformed into a fixed-time stable one through scaling, thereby reinforcing the hypothesis that all convergence rates are interconnected.

Original authors: Kunal Garg

Published 2026-03-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kunal Garg

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The Race to the Finish Line

Imagine you are driving a car toward a destination (let's call it "Home," which represents the equilibrium point or the state of perfect stability).

For decades, engineers and mathematicians have studied how fast a car can get home. They have different categories for this:

Asymptotic Stability: You get closer and closer to home, but theoretically, you never quite arrive. It's like a runner who slows down as they approach the finish line, getting infinitely close but taking forever to cross it.
Exponential Stability: You get home very fast, like a rocket. The distance shrinks rapidly, but you still technically take infinite time to hit the exact zero point.
Finite-Time Stability: You arrive at home in a specific, calculable amount of time (e.g., "I will be there in exactly 10 minutes").
Fixed-Time Stability: This is the "superpower." No matter where you start—whether you are 1 mile away or 1,000 miles away—you are guaranteed to arrive within a specific maximum time limit (e.g., "I will be there in 10 minutes, no matter what").

The Problem:
Usually, if a system is designed to be "Asymptotically Stable" (slow but steady), it is very hard to make it "Fixed-Time Stable" (fast and guaranteed) without completely redesigning the engine. Most existing math tools required a very specific, smooth "map" (a differentiable Lyapunov function) to prove these fast speeds were possible. If the map was bumpy or broken (non-smooth), the old tools failed.

The Solution (The Paper's Contribution):
Kunal Garg's paper says: "Stop worrying about the map. Just change the speed of the car."

The author proves that any system that eventually gets home (Asymptotic Stability) can be transformed into a system that gets home in a guaranteed, fixed time (Fixed-Time Stability). You don't need a perfect, smooth map to prove this; you just need to look at how the car behaves near the finish line.

The Key Analogies

1. The "Speed Limit" Transformation

Imagine you have a car that drives toward a stop sign.

The Old Way: To make the car stop instantly, you had to build a brand-new car with a special engine.
The New Way (This Paper): You keep the exact same car, but you install a smart speed governor.
- If the car is far away, the governor makes it go super fast.
- If the car is very close to the stop sign, the governor makes it go even faster (counter-intuitively, to prevent it from slowing down too early).
- If the car is in the middle, it goes at a normal speed.

The paper provides the mathematical "rulebook" for this governor. It shows that by simply scaling the speed based on how far you are from the goal, you can turn a "slowly arriving" system into a "guaranteed arrival" system.

2. The "Bumpy Road" (Non-Smooth Analysis)

Imagine driving on a road with potholes (non-smooth dynamics).

Old Tools: The old math tools were like a delicate glass ruler. If you tried to measure the road with potholes, the ruler would break. You needed a perfectly smooth road to use them.
New Tools: This paper introduces a rubber tape measure. It can stretch and bend over the potholes. It allows the author to prove that even if the road is bumpy (the system is non-smooth), the car will still reach the destination in a fixed time if you apply the right speed governor.

3. The "Universal Translator"

The paper argues that Asymptotic, Finite-Time, and Fixed-Time stability are actually the same thing, just spoken in different languages.

Think of them as different dialects of the same language.
The author found the "translator." If you have a system that speaks "Asymptotic" (slowly converging), you can translate it into "Fixed-Time" (fast converging) by applying a specific mathematical transformation (scaling).
This is huge because it means you don't need to invent a new system for every problem. You can take a standard, stable system and "translate" it to be super-fast.

Why Does This Matter?

In the real world, we want things to happen fast and reliably.

Robotics: If a robot arm needs to grab a moving object, it can't afford to "slowly approach" the object. It needs to arrive at the exact right spot at the exact right time.
Machine Learning: When training AI, we want the computer to find the best answer quickly. If the math says "it will eventually get there," that's too slow. We want "it will get there in 5 minutes."
Robustness: The paper mentions that faster convergence makes systems more resistant to "noise" or "disturbances." Think of it like a tightrope walker. If they correct their balance slowly (Asymptotic), a sudden gust of wind might knock them off. If they correct their balance instantly (Fixed-Time), they can shrug off the wind and stay on the wire.

The "Magic" Formula

The paper essentially says:

"If you have a system that is stable (it won't crash), you can multiply its speed by a specific factor (based on how far it is from the goal) to make it arrive in a fixed time, regardless of where it started."

It removes the need for perfect, smooth mathematical descriptions of the system, allowing engineers to apply these "super-fast" guarantees to messy, real-world problems like optimizing neural networks or controlling complex robots.

Summary

This paper is like discovering a universal turbo-boost button. It proves that if a machine is safe and stable, you can press this button (using the author's new math rules) to make it finish its job in a guaranteed, fixed amount of time, even if the machine is messy or bumpy. It connects all the different ways we measure "speed to stability" and shows they are all part of the same family.

1. Problem Statement

The paper addresses a fundamental theoretical question in control theory and optimization: What is the relationship between different convergence rates? Specifically, it investigates the equivalence between:

Asymptotic Stability (AS): Convergence as $t \to \infty$ .
Exponential Stability (ES): Convergence at an exponential rate.
Finite-Time Stability (FTS): Convergence to an equilibrium in finite time $T(x_0)$ .
Fixed-Time Stability (FxTS): Convergence in finite time where the settling time $T(x_0)$ is uniformly bounded for all initial conditions.

While previous works established equivalences for specific classes of systems (e.g., gradient flows) or required the existence of a differentiable, positive-definite Lyapunov function, this paper aims to:

Relax the requirement for a differentiable Lyapunov function by utilizing non-smooth analysis and LaSalle-like invariance principles.
Prove that any dynamical system with a globally asymptotically stable (GAS) equilibrium can be transformed (via scaling/normalization) into a system with a Fixed-Time Stable equilibrium.
Establish a unified framework showing that AS, ES, FTS, and FxTS are interconnected through suitable geometric transformations of system dynamics.

2. Methodology

The author employs a combination of non-smooth analysis and invariance principles to derive new stability conditions.

Non-Smooth Lyapunov Analysis: Instead of requiring a continuously differentiable Lyapunov function $V$ , the paper utilizes locally Lipschitz functions. It employs the Lie derivative for non-smooth functions (defined as a set-valued map $\tilde{L}_f V$ ) and utilizes Filippov solutions for discontinuous dynamics.
LaSalle-like Invariance Principles: The core theoretical tool is an extension of LaSalle's invariance principle. The paper establishes conditions under which trajectories converge to the largest invariant set where the derivative of a function is zero, even without the function being positive definite.
Temporal Transformation (Scaling): The paper introduces a method to modify system dynamics $\dot{x} = f(x)$ by scaling the vector field $f(x)$ with powers of its norm $\|f(x)\|$ . This effectively changes the time scale of the system, accelerating convergence.
Comparison Lemma: The convergence time is analyzed by comparing the system's Lyapunov-like function evolution to a scalar differential equation $\dot{v} = -\phi(v)$ , where the integral of $1/\phi(v)$ determines the settling time.

3. Key Contributions

A. New Non-Smooth Stability Theorems

The paper presents Theorem 2 and Theorem 3, which provide first and second-order non-smooth conditions for FTS and FxTS:

Theorem 2: Establishes FTS/FxTS using a locally Lipschitz function $V$ (not necessarily positive definite) and a scalar function $\phi$ . It proves convergence to an invariant set $E$ within a time bounded by $\int \frac{dV}{\phi(V)}$ .
Theorem 3: Extends this to second-order derivatives (using the Lie derivative of the Lie derivative), allowing for conditions based on the dynamics of $\dot{V}$ rather than just $V$ . This is crucial for systems where $V$ is not strictly decreasing but its rate of change is.

B. Equivalence via System Scaling

The paper demonstrates that Global Asymptotic Stability (GAS) implies Fixed-Time Stability (FxTS) under a specific transformation:

Theorem 4: Shows that if a system $\dot{x} = f(x)$ is GAS, the modified system:
$\dot{x} = \left( \frac{1}{\|f(x)\|^p} + \frac{1}{\|f(x)\|^q} \right) f(x)$
(where $0 < p < 1$ and $q < 0$ ) is FxTS.
Theorem 5: Applies this logic to gradient flows ( $\dot{x} = -\nabla V$ ), proving that a piecewise scaling of the gradient field results in FxTS.

C. Generalization Beyond Gradient Flows

Unlike previous works limited to gradient flows or convex optimization, these results apply to a general class of nonlinear dynamical systems, provided the equilibrium is GAS and specific matrix conditions (related to the Hessian-like matrix $H(x)$ ) are met.

4. Key Results

Relaxed Lyapunov Conditions: The paper proves that FTS and FxTS can be guaranteed without a differentiable, positive-definite Lyapunov function. It suffices to have a locally Lipschitz function whose derivative is negative definite (or satisfies specific bounds) on the system trajectories.
Unified Equivalence: The paper formally proves that AS $\iff$ FTS $\iff$ FxTS under suitable transformations. Specifically, any GAS system can be "time-scaled" to become FxTS.
Chattering-Free Continuous Dynamics: The proposed scaling methods allow for continuous dynamics (unlike sliding mode control which often induces chattering), making them applicable to a broader range of physical and optimization problems.
Settling Time Bounds: Explicit upper bounds for the settling time $T$ are derived in terms of the system parameters and the initial conditions, showing that $T$ is bounded globally (FxTS) when the scaling exponents satisfy specific conditions ( $p < 1, q < 0$ ).

5. Significance and Impact

Theoretical Unification: The work bridges the gap between classical asymptotic stability and modern finite/fixed-time stability, suggesting they are not distinct categories but rather different representations of the same underlying stability properties under time transformation.
Robustness and Optimization: Since faster convergence rates (FTS/FxTS) offer improved robustness against disturbances, these results provide a theoretical foundation for designing more robust controllers and faster optimization algorithms (e.g., for neural network training).
Broader Applicability: By removing the strict requirement for differentiable Lyapunov functions and extending results beyond gradient flows, the methodology can be applied to non-smooth optimization problems and complex non-linear control systems where traditional Lyapunov analysis fails.
Future Directions: The author highlights the potential for applying these results to non-smooth optimization algorithms and the design of control systems that require guaranteed convergence times.

In summary, Kunal Garg's paper provides a rigorous mathematical framework demonstrating that asymptotic stability can be transformed into fixed-time stability through dynamic scaling, utilizing non-smooth invariance principles to generalize these findings beyond previous limitations.