Stability equibrilium solution

[dirk_mec1]

Homework Statement

Determine the stability property of the following equibrilium solution:

(0,0) of

$$\ddot{x}+ \alpha x +x=0$$ with $$\alpha \in \mathbb{R}$$

Homework Equations

- A solution is Lyapunov stable if for each $\epsilon$ there is a $\delta$ such that: $$||x(0)|| \leq \delta$$ yields $$||x(t)|| \leq \epsilon$$

- A solution is unstable if it isn't stable.

- A solution is asymptotically stable if there is a delta such that: $$||x(0)|| \leq \delta$$ yields $$\lim_{t \rightarrow \infty} ||x(t)|| = 0$$

Note: x can be a vector.

The Attempt at a Solution

I don't know how to use these defintions. Can I just use the 2-norm? Can someone explain me how this works?

 

[Nino MMP]

The equilibrium solution (0,0) for this differential equation is asymptotically stable. To show this, we need to show that for any \epsilon > 0, there exists a \delta > 0 such that ||x(0)|| \leq \delta implies ||x(t)|| \leq \epsilon for all t \geq 0. Since the equilibrium solution is (0,0), this means that any initial condition with ||x(0)|| \leq \delta will converge to (0,0). This means that the solution is asymptotically stable.