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Asymptotic stability of a system ( ordinary DE)

  1. Dec 7, 2005 #1
    Determine the asymptotic stability of the system x' = Ax where

    A is 3 x 3 matrix

    A = -1 1 1
    0 0 1
    0 0 -2

    ( first row is -1 1 1 second is 0 0 1 and third is 0 0 -2)

    More specifically, what stability conclusion(s) can be drawn? ( Justify your answer)
  2. jcsd
  3. Dec 7, 2005 #2

    D H

    Staff: Mentor

    The solution of this system is
    [tex] x(t) = exp(At) x(t=0) [/tex]
    Try evaluating
    [tex] exp(At) = \sum_n \frac{t^n}{n!}A^n [/tex]
    This matrix holds the answers to your question.
  4. Dec 8, 2005 #3


    User Avatar
    Science Advisor
    Homework Helper

    I think analyzing this system qualitatively along the lines presented in "Differential Equations" by Blanchard, Devaney, and Hall (other books too) is a nice way of drawing conclusions about this and other systems. Try it.:smile:

    Of course, first do a few 2-D ones.
  5. Dec 8, 2005 #4


    User Avatar
    Staff Emeritus
    Science Advisor

    Have you determined the eigenvalues of that matrix?
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