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Homework Help: Stability equibrilium solution

  1. Dec 21, 2008 #1
    1. The problem statement, all variables and given/known data
    Determine the stability property of the following equibrilium solution:

    (0,0) of

    [tex] \ddot{x}+ \alpha x +x=0 [/tex] with [tex] \alpha \in \mathbb{R} [/tex]

    2. Relevant equations
    - A solution is Lyapunov stable if for each [itex] \epsilon[/itex] there is a [itex] \delta [/itex] such that: [tex]||x(0)|| \leq \delta[/tex] yields [tex]||x(t)|| \leq \epsilon [/tex]

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

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

    Note: x can be a vector.

    3. 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?
     
  2. jcsd
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