The question reads: For which real numbers k is the zero state a stable equilibrium of the dynamic system x t + 1 = A x t ? A = [ 0.1 k 0.3 0.3 ] So, my thought is I need to find the eigenvalues. In order to do this I calculated the characteristic polynomial as x 2 − 0.4 x + 0.03 − 0.3 k = 0, with x representing eigenvalues.Using the quadratic formula I found that the (real) eigenvalues are x = 2 ± 1 + 30 k 10 and for the zero state to be in stable equilibrium 1 + 30 k &lt; 8. Hence, k &lt; 21 / 10 (for stable equilibrium).My question is how do I figure out the values for k if the eigenvalues are complex?Do I solve the inequality − 1 − 30 k &lt; 8?

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The question reads: For which real numbers k is the zero state a stable equilibrium of the dynamic system       x          t      +      1        =  A      x    t  ?  A  =      [                            0.1                          k                                      0.3                          0.3                      ]  So, my thought is I need to find the eigenvalues. In order to do this I calculated the characteristic polynomial as      x    2    −  0.4  x  +  0.03  −  0.3  k  =  0, with   x representing eigenvalues.Using the quadratic formula I found that the (real) eigenvalues are  x  =            2      ±              1        +        30        k              10  and for the zero state to be in stable equilibrium       1    +    30    k    &amp;lt;  8. Hence,   k  &amp;lt;  21  /  10 (for stable equilibrium).My question is how do I figure out the values for   k if the eigenvalues are complex?Do I solve the inequality       −    1    −    30    k    &amp;lt;  8?

frethi38 · Accepted Answer

In general, the zero state of a dynamical system of the form       x          t      +      1        =  A      x    t   is a stable equilibrium when all of the eigenvalues of A are inside the unit circle in the complex plane. So if you have an eigenvalue of the form   a  +  b  i, you get stability when       a    2    +      b    2    &amp;lt;  1. Since this is a homework question, I&#039;ll let you work out the rest of the problem from here. (This isn&#039;t going to be equivalent to solving       −    1    −    30    k    &amp;lt;  8, though.)

George Bray · Accepted Answer

Good answer. Just to be clear, though, one has asymptotic stability if       a    2    +      b    2    &amp;lt;  1. The system is generally considered stable if       a    2    +      b    2    =  1.

The question reads: For which real numbers k is the zero state a stable equilibrium of the dynamic

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