Results 1 to 2 of 2

Thread: Lyapunov function-Differential equation prob

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    78

    Lyapunov function-Differential equation prob

    For dx/dt = y^3 - 2*x^3

    dy/dt = -3*x - y^3

    Show that the origin is nonlinearly stable by nding the appropriate Lyapunov function.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Hey, I don't have it but let me try to rough it in ok:

    So we have:

    $\displaystyle x'=y^3-2x^3$

    $\displaystyle y'=-3x-y^3$

    In these, sometimes we can find a Lyapunov function of the form $\displaystyle ax^2+by^2$ for suitable choices of a and b such that $\displaystyle L(p_0)=0$ at the critical point $\displaystyle p_0$, $\displaystyle L(x)>0$ if $\displaystyle x\neq p_0$ in some region about $\displaystyle p_0$ and $\displaystyle F\cdot\nabla L \leq 0$ in that same region. If so, then the point $\displaystyle p_0$ is stable. Well the critical point is the origin so let $\displaystyle a>0$ and $\displaystyle b>0$ and then:

    $\displaystyle F\cdot\nabla L =(y^3-2x^3,-3x-y^3)\cdot(2ax,2bx)=2axy^3-4ax^4-6bxy-2by^4$

    $\displaystyle =2axy^3-6bxy-(4ax^4+2by^4)$

    The last term is always negative. However, I can't show that for suitable choices of a and b, the entire expression is less than or equal zero in some deleted neighborhood of the origin.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lyapunov equation
    Posted in the Calculators Forum
    Replies: 1
    Last Post: May 13th 2009, 07:29 AM
  2. Finite string Partial differential equation prob
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: Feb 23rd 2009, 04:38 PM
  3. Lyapunov equation and eigenvalues
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Dec 11th 2008, 08:53 PM
  4. Lyapunov Differential Equation
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Nov 7th 2008, 06:31 AM
  5. HELP!! differential equation and bessels function
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: Jun 14th 2008, 09:35 AM

Search Tags


/mathhelpforum @mathhelpforum