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Lie derivatives and Lipschitz contants

  1. Jan 29, 2013 #1
    Hello all,

    I have a problem with an inequality.
    Let

    \dot%20x=f(x).gif

    Is the following proof valid?

    20%20\frac{\partial%20f}{\partial%20x}%20\dot%20x%20=%20\frac{\partial%20f}{\partial%20x}%20f(x).gif

    from which, taking the norm to both sides yields

    f(x)%20||%20\le%20||\frac{\partial%20f}{\partial%20x}%20||%20||f(x)%20||%20\le%20L%20||f(x)%20||.gif

    where L is the Lipschitz constant of f w.r.t. x.
    Thus, can I conclude that


    %20||\frac{d}{dt}%20f(x)%20||%20\le%20L%20||%20f(x)%20||.gif

    Is it correct?

    Thanks :)
     
  2. jcsd
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