# Stable by but not Asymptotically Stable

• Sep 8th 2010, 09:51 AM
joestevens
Stable by but not Asymptotically Stable
Show that the equilibrium point $\displaystyle x_0=0$ for the differential equation $\displaystyle x^{\prime}=0$ is stable but not asymptotically stable.
• Sep 8th 2010, 05:29 PM
JJMC89
Quote:

Originally Posted by joestevens
Show that the equilibrium point $\displaystyle x_0=0$ for the differential equation $\displaystyle x^{\prime}=0$ is stable but not asymptotically stable.

I don't know exactly how to do this but you probably need to use the ε, δ definition of stable and asymptotically stable.
• Sep 8th 2010, 05:30 PM
joestevens
Quote:

Originally Posted by JJMC89
I don't know exactly how to do this but you probably need to use the ε, δ definition of stable and asymptotically stable.

I got as far as knowing that but I don't know where to do. Anyone have any suggestions as to how to do this?
• Sep 9th 2010, 04:26 AM
Jester
In this problem you can actually integrate and find the solution

$\displaystyle x(t) = x_0$

Now, with $\displaystyle x_0 = 0$ gives your solution. Now perturb it so

$\displaystyle x^{*}(t) = x_1$.

So,

$\displaystyle | x(t) - x^{*}(t)| = x_1$ for all t (this a stable)

However, we don't have for some $\displaystyle \delta > 0$

$\displaystyle | x(t) - x^{*}(t)| \le \delta \; \text{for}\; t > t*$ (this a asymptotic stability).