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Lyapunov function help!

  1. Nov 27, 2012 #1
    Assume that f(0) = 0 and Df(0) has eigenvalues with negative real parts. Con-
    struct a Lyapunov function to show that 0 is asymptotically stable.
  2. jcsd
  3. Nov 27, 2012 #2
    I know it is a strict lyapunov function, but i cannot figure out how to solve it for the general case.
  4. Dec 5, 2012 #3


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    Homework Helper

    Consider [itex]V(x) = \|x\|^2 = x \cdot x[/itex]. Then [itex]\nabla V = 2x[/itex] and
    \dot V = \nabla V \cdot \dot x = \nabla V \cdot f(x) =
    2x \cdot (Df(0) \cdot x) + O(\|x\|^3).
    What does the condition on the eigenvalues of Df(0) imply about the sign of [itex]x \cdot (Df(0) \cdot x)[/itex]? What does that imply about [itex]\dot V[/itex] for [itex]\|x\|[/itex] sufficiently small?

    Can you prove that if [itex]\dot V < 0[/itex] on a neighbourhood of 0 then 0 is asymptotically stable?
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