Thread: Stable by but not Asymptotically Stable

1. Stable by but not Asymptotically Stable

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

2. Originally Posted by joestevens
Show that the equilibrium point $x_0=0$ for the differential equation $x^{\prime}=0$ 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.

3. 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?

4. In this problem you can actually integrate and find the solution

$x(t) = x_0$

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

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

So,

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

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

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