- #1

joypav

- 151

- 0

$$p(z)=det(zI-A),$$

where I is the n x n identity matrix.

Show that any solution of the system

$$\dot{x}=Ax+b$$

remains bounded as $t\to{\infty}$.

Work out in detail the solution of the system with

$$

\begin{bmatrix}

0 & w \\

-w & 0 \\

\end{bmatrix}, w>0

$$

to verify what happens in one special case.

So we have...

$$p(z)=det(zI-A)$$

$$p(z)=det( \begin{bmatrix}

z & 0 \\

0 & z \\

\end{bmatrix}-

\begin{bmatrix}

0 & w \\

-w & 0 \\

\end{bmatrix})=

det(\begin{bmatrix}

z & -w \\

w & z \\

\end{bmatrix})=z^2+w^2=(z+wi)(z-wi)

$$

Both zeros are on the imaginary axis and are both of order one.

Now I am supposed to find the solution of the system?

Do we use that the solution of the system is of the form

$$x_l(t)=\sum_{j=1}^2p_{jl}(t)e^{\lambda_jt}, l=1,2?$$

So

$$\lambda_1=wi,\lambda_2=-wi$$

I have no idea if I'm going in the right direction or not.

I would really like some help with the specific problem, not necessarily the more generalized proof. I think if I knew how to solve the specific problem given then I'd have a much better understanding.