Math 6410 • ODE

\textbf{Stability of Solutions.}
The following questions have to do with the stability of the zero solution \(z(t)=0\) to a
system of ODE's
\[
  y'=f(t,y),
\]
where \(f(t,x)\in C^\infty(\R\times\R^n,\R^n)\) such that \(f(t,0)=0\) for all \(t\).
\begin{enumerate}
\item[(a.)] Define: \(z\) is Liapunov Stable. Define: \(z\) is asymptotically stable.
\item[(b.)] Give an example of an ODE system such that the zero solution is Liapunov
stable but not asymptotically stable.
\item[(c.)] Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda_i\) satisfy
\(\operatorname{Re}\lambda_i<0\) for all \(i=1,\ldots,n\). Let
\(g(x)\in C^\infty(\R^n,\R^n)\) such that there are constants \(1<\beta,c<\infty\) so that
\(\lvert g(x)\rvert\le c\lvert x\rvert^\beta\) for all \(x\in\R^n\). Show that the zero
solution \(z(t)\) is asymptotically stable in the equation
\[
  y'=Ay+\sin(t)g(y).
\]
\end{enumerate}