Math 6410 • ODE

Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda_i\) satisfy
\(\Re\lambda_i<0\) for all \(i=1,\ldots,n\). Answer the questions about the linear system
\begin{equation}
  \begin{cases}
    \dfrac{dx}{dt}=Ax,\\
    x(0)=x_0.
  \end{cases}
  \tag{1}
\end{equation}
\begin{enumerate}
\item[(a)] Define: \(V(x)\) is a \textit{Liapunov Function}.
\item[(b)] Find a Liapunov Function.
\item[(c)] Define: \(z(t)=0\) is an \textit{Asymptotically Stable} solution for (1).
\item[(d)] Using your Liapunov Function, show that \(z(t)=0\) is an asymptotically stable
solution.
\end{enumerate}