Linear Systems equillibrium

  • Thread starter Mynock
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  • #1
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Hey, I'm trying to find two linear functions, f(r) and h(r) for the following system:

dp/dt = A*f(r)

dr/dt = -B*h(r)

where A and B are constants greater than zero. I'm trying to find linear functions that will tend to equillibrium, and also where

limit df(r)/dt = 0
r->0

I have been trying various linear functions and have been unable to come up with a solution. Is there a solution? and if so, what would be one? It's probalby something simple that I'm overlooking. Any help would be appreciated. Thanks.
 

Answers and Replies

  • #2
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The linear (homogeneous) functions in one variable, 'r' in this case, have the general form:

[tex]f(r) \ = \ \alpha\cdot r,\quad g(r) \ = \ \beta\cdot r, \qquad \alpha,\beta\in\mathbb{R}[/tex]

So this system can be trivially solved, since the variables are separated. On one hand, you have:

[tex]\dot{r}\ = \ -B\beta\, r[/tex]

so

[tex]r(t) \ = \ r_0e^{-B\beta t}[/tex]

The condition for stability is obviously [tex]\beta > 0[/tex].

On the other hand:

[tex]\dot{p}\quad = \quad A\alpha\, r \quad = \quad A\alpha r_0 e^{-B\beta t}[/tex]

Then,

[tex]p(t) \quad = \quad \gamma \ - \ \frac{A\alpha r_0}{B\beta} e^{-B\beta t}[/tex]

where [tex]\gamma[/tex] is an integration constant. This means that, as long as [tex]\beta > 0[/tex], your system will be stable, with equilibrium point [tex]p = \gamma, r = 0[/tex].
 

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