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Generalization of Comparison Theorem

  1. Oct 7, 2013 #1
    I was wondering if there is a generalization of the following (roughly stated) theorem to n-dimensional systems:

    Given some restrictions on the functions [itex]f[/itex] and [itex]g \in \Re[/itex], let [itex]y_s(t)[/itex] and [itex]x_s(t) \in \Re[/itex] be solutions to the initial value problems:


    [tex]
    \dot{x}(t) = f(x,t), x(t_0) = x_0\\
    \dot{y}(t) = g(x,t), y(t_0) = y_0
    [/tex]

    Then if [itex]f(x,t) > g(x,t)[/itex] in some domain [itex]D[/itex] containing [itex]x_0[/itex] and [itex]y_0[/itex] and [itex]x_0 \geq y_0[/itex], [itex]x_s(t) > y_s(t) \forall t > t_0[/itex] when the trajectories stay in this domain [itex]D[/itex].

    This is a rough statement of the theorem, but I was wondering: is there an analogous statement for n-dimensional systems? For instance, is the following true?

    Given some restrictions on the functions [itex]f[/itex] and [itex]g \in \Re^n[/itex], let [itex]x_s(t)[/itex] and [itex]y_{s}(t) \in \Re^n[/itex] be solutions to the initial value problems:


    [tex]
    \dot{x}(t) = f(x,t), x(t_0) = x_0\\
    \dot{y}_{1}(t) = g(x,t), y(t_0) = y_{0}
    [/tex]

    Then if [itex]f(x,t)[/itex] is in the convex hull of [itex]g(x,t)[/itex] in some domain [itex]D[/itex] containing [itex]x_0[/itex] and [itex]y_0[/itex] and [itex]x_0[/itex] is in the convex hull of [itex]y_0[/itex], [itex]x_s(t)[/itex] is in the convex hull of [itex]y_s(t) \forall t > t_0[/itex] when the trajectories stay in this domain [itex]D[/itex].

    This result seems like it should be true, intuitively, but I'm having trouble finding a formal statement of it.
     
    Last edited: Oct 7, 2013
  2. jcsd
  3. Oct 7, 2013 #2
    Sorry -- I was a little hasty in writing this and meant to say the functions [itex]f[/itex] and [itex]g[/itex] map [itex]\Re^2[/itex] to [itex]\Re[/itex], or, in the second possible theorem, map [itex]\Re^{n+1}[/itex] to [itex]\Re^n[/itex].
     
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