Math 6410 • ODE

Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda_i\) satisfy
\(\Re\lambda_i<0\) for all \(i=1,\ldots,n\). Consider the initial value problem for
\(x_0\in\R^n\),
\begin{equation}
  \begin{cases}
    \dfrac{dx}{dt}=Ax+h(x,t), & \text{with }
    h(x,t)=\dfrac{t\norm{x}^3}{1+t}v,\\
    x(0)=x_0
  \end{cases}
  \tag{2}
\end{equation}
where \(v\in\R^n\) is a unit vector.
\begin{enumerate}
\item[(a)] Show that if \(\lvert x_0\rvert\) is small enough, there is a bounded solution
to (2) that exists for all time.

[Hint: Using variation-of-parameters and the fact that
\[
  \lim_{\norm{x}\to0}\left(\norm{h(x,t)}/\norm{x}\right)=0
\]
uniformly in \(t\), for \(t>0\), obtain an integral inequality for \(\norm{x}\) and then
apply Gronwall's Inequality: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions and
\(c_0\ge0\) is a constant that satisfy
\(u(t)\le c_0+\int_0^t g(s)u(s)\dd s\) for all \(t\ge0\). Then Gronwall's Inequality implies
\[
  u(t)\le c_0\exp\left(\int_0^t g(s)\dd s\right)
\]
for all \(t\ge0\).]
\item[(b)] Let \(z(t)=0\) for all \(t\) be the zero solution to (2). Define what it means for
\(z\) to be \textit{Lyapunov Stable}. Show that \(z\) is Lyapunov stable for this equation.
\end{enumerate}