Asymptotic stability of a system ( ordinary DE)

[ODEMath]

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)

 

[D H]

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
$$x(t) = exp(At) x(t=0)$$
Try evaluating
$$exp(At) = \sum_n \frac{t^n}{n!}A^n$$
This matrix holds the answers to your question.

 

[saltydog]

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.

Of course, first do a few 2-D ones.

 

[HallsofIvy]

Have you determined the eigenvalues of that matrix?