---
tag: VY180V8QY2
kind: concept
name: Exponential stability
msc: [34D20, 93D20]
specializes: [0NB689782W]
sources:
  - url: https://en.wikipedia.org/wiki/Exponential_stability
same_as:
  wikidata: Q17011487
  wikipedia: Exponential stability
---

## Summary

An equilibrium is exponentially stable when nearby solutions approach it at least as fast as a decaying exponential.
It is a strong form of [[asymptotic stability|0NB689782W]] that comes with an explicit rate.
For a linear system $x' = Ax$, asymptotic and exponential stability coincide, and both mean that every eigenvalue of $A$ has negative real part.
Exponential stability survives small perturbations of the system, which makes it the preferred notion in control theory.
See the [[definition|V6E2R7JNN8]].

## Explanations

### L1 human

A solution that is disturbed from rest returns to rest, and the size of the disturbance shrinks by a fixed factor in each unit of time.
For $x' = -x$, the solution $x_0 e^{-t}$ loses about two thirds of its size every unit of time.
For $x' = -x^3$, solutions also tend to $0$, but only like $1/\sqrt{2t}$, which is slower than any exponential.
So the origin of $x' = -x^3$ is asymptotically stable but not exponentially stable.