Hi,

Using these definitions:

A homogeneous equation with constant coefficients is said to be stable if all solutions remain bounded as t-->infinity

A homogeneous equation with constant coefficients is said to be asymptotically stable if all solutions converge to the zero solution as t-->infinity

I'm trying to prove these two statements:

1) The system is stable if and only if for every root $\displaystyle \lambda = \alpha + i\beta$ of the characteristic polynomial, $\displaystyle 0 \geq \alpha $ and $\displaystyle \alpha < 0 $ if the multiplicity of lambda is bigger than one.

2) The system is aymptotically stable if and only if for every root $\displaystyle \lambda = \alpha + i\beta$ of the characteristic polynomial, $\displaystyle \alpha < 0 $.

Thanks a lot!