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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.

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.

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