Math 6410 • ODE

\textbf{Constant Coefficients.}
Suppose that \(A\) is a real \(n\times n\) matrix all of whose eigenvalues \(\lambda\) have
strictly negative real part \(\operatorname{Re}\lambda<0\). Suppose
\[
  \frac{dx}{dt}=Ax,\qquad x(0)=x_0. \tag{1.}
\]
\begin{enumerate}
\item[(a.)] Show that there is a decreasing function \(\delta(t)>0\) such that
\(\delta(t)\to0\) as \(t\to\infty\) and such that every solution of (1.) satisfies
\[
  \lvert x(t)\rvert\le\delta(t)\lvert x_0\rvert
  \qquad\text{as }t\to\infty.
\]
\item[(b.)] Define: \(z\) is Liapunov Stable. Define: \(z\) is Asymptotically Stable. Show
that the zero solution \(z(t)=0\) of the autonomous system is asymptotically stable.
\end{enumerate}