91Ó°ÊÓ

Consider the linear system $\frac{d\stackrel{→}{x}}{dt}=A\stackrel{→}{x}$, where A is the real $2×2$matrix with complex eigenvalues$pÂ\pm iq$ and $q≠0$.

Consider an eigenvector $\stackrel{→}{v}+i\stackrel{→}{w}$with eigenvalue$pÂ\pm iq$ . Then:

$\stackrel{→}{x}\left(t\right)=e^{pt}S\left(\begin{array}{cc}\cos \left(qt\right)& -\sin \left(qt\right)\\ \sin \left(qt\right)& \cos \left(qt\right)\end{array}\right)S^{-1}{\stackrel{→}{x}}_0$, where $S=\left[\stackrel{→}{w}   \stackrel{→}{v}\right]$

Recall that $S^{-1}{\stackrel{→}{x}}_0$is the coordinate vector of $x_0$with respect to the basis$\stackrel{→}{w},\stackrel{→}{v}$

For a system, $\frac{d\stackrel{→}{x}}{dt}=A\stackrel{→}{x}$here A is the matrix form.

The zero state is an asymptotically stable equilibrium solution if and only if the real parts of all eigen values of A are negative.

Consider the stability of the system is$\frac{d\stackrel{→}{x}}{dt}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ -1& -1& -2\end{array}\right)\stackrel{→}{x}$

Here A is the real matrix:

$A=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -1& -2\end{array}\right)$

$$\begin{gathered} \frac{d\stackrel{→}{x}}{dt}=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -1& -2\end{array}\right)\stackrel{→}{x} \\ A=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -1& -2\end{array}\right) \end{gathered}$$

Find $A-\mathrm{λ}I$as:

$\begin{array}{rcl}A-\mathrm{λ}I& =& 0\\ \left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -1& -1\end{array}\right)-\mathrm{λ}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)& =& 0\\ \left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -1& -1\end{array}\right)-\left(\begin{array}{ccc}\mathrm{λ}& 0& 0\\ 0& \mathrm{λ}& 0\\ 0& 0& \mathrm{λ}\end{array}\right)& =& 0\\ \left(\begin{array}{ccc}-\mathrm{λ}& 1& 0\\ 0& -\mathrm{λ}& 1\\ -1& -1& -2-\mathrm{λ}\end{array}\right)& =& 0\end{array}$

Find characteristics polynomial as:

$\begin{array}{rcl}-\mathrm{λ}\left[\left(-\mathrm{λ}\right)\left(-2-\mathrm{λ}\right)-\left(-1\right)\right]-1\left(0-\left(-1\right)\right)& =& 0\\ -\mathrm{λ}(2\mathrm{λ}+{\mathrm{λ}}^2+1)-\left(1\right)& =& 0\\ -2{\mathrm{λ}}^2-{\mathrm{λ}}^3-\mathrm{λ}-1& =& 0\\ {\mathrm{λ}}^3+2{\mathrm{λ}}^2+\mathrm{λ}+1& =& 0\\ & & \mathrm{λ}   is  negative\\ & & \\ & & \end{array}$

The stability of the system is negative for all eigen values.

Hence the system is asymptotically stable.