Asymptotic stability of a system ( ordinary DE)

In summary, the system x' = Ax is asymptotically stable if the eigenvalues of A have negative real parts. In this case, the eigenvalues are -1, -2, and -2, all of which have negative real parts. Therefore, the system is asymptotically stable. This conclusion can be justified by analyzing the system qualitatively using the methods presented in "Differential Equations" by Blanchard, Devaney, and Hall.
  • #1
ODEMath
4
0
Determine the asymptotic stability of the system x' = Ax where

A is 3 x 3 matrix

A = -1 1 1
0 0 1
0 0 -2

( first row is -1 1 1 second is 0 0 1 and third is 0 0 -2)

More specifically, what stability conclusion(s) can be drawn? ( Justify your answer)
 
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  • #2
ODEMath said:
Determine the asymptotic stability of the system x' = Ax where
A is 3 x 3 matrix
A = -1 1 1
0 0 1
0 0 -2
( first row is -1 1 1 second is 0 0 1 and third is 0 0 -2)
More specifically, what stability conclusion(s) can be drawn? ( Justify your answer)

The solution of this system is
[tex] x(t) = exp(At) x(t=0) [/tex]
Try evaluating
[tex] exp(At) = \sum_n \frac{t^n}{n!}A^n [/tex]
This matrix holds the answers to your question.
 
  • #3
ODEMath said:
More specifically, what stability conclusion(s) can be drawn? ( Justify your answer)

I think analyzing this system qualitatively along the lines presented in "Differential Equations" by Blanchard, Devaney, and Hall (other books too) is a nice way of drawing conclusions about this and other systems. Try it.:smile:

Of course, first do a few 2-D ones.
 
  • #4
Have you determined the eigenvalues of that matrix?
 

1. What is asymptotic stability of a system?

Asymptotic stability of a system refers to the behavior of a system over time, specifically whether the system's solutions approach a constant value as time goes to infinity.

2. How is asymptotic stability determined for a system?

Asymptotic stability can be determined by analyzing the eigenvalues of the system's Jacobian matrix at critical points. If all eigenvalues have negative real parts, the system is asymptotically stable.

3. What is the difference between asymptotic stability and stability?

Stability refers to a system's behavior over time, while asymptotic stability specifically refers to the behavior of a system's solutions as time goes to infinity. A system can be stable but not asymptotically stable if its solutions approach a limit that is not a constant value.

4. Can a system be asymptotically stable for some initial conditions but not others?

Yes, a system can be asymptotically stable for some initial conditions but not others. This is because the behavior of a system's solutions can vary depending on the initial conditions.

5. How is asymptotic stability important in practical applications?

Asymptotic stability is important in practical applications because it allows us to determine the behavior of a system over time and make predictions about its long-term behavior. This can be useful in fields such as engineering, economics, and biology.

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