Math 6410 • ODE

Consider the following linear equation for \(x\in\R^n\):
\[
  \dot{x}=A+B(t)x,\qquad x(t_0)=x_0.
\]
where \(A,B(t)\) are \(n\times n\) matrices. Suppose that all eigenvalues \(\lambda_j\),
\(j=1,\ldots,n\), of the matrix \(A\) satisfy \(\Reop(\lambda_j)<0\), and let \(B(t)\)
be continuous for \(0\le t<\infty\) with
\(\int_0^\infty\norm{B(t)}\dd t<\infty\).
\begin{enumerate}
\item[(a)] Using the variation of constants formula show that there exist constants
\(K,\sigma>0\) such that
\[
  \abs{x(t)}
  \le Ke^{-\sigma(t-t_0)}\abs{x_0}
  +K\int_{t_0}^t e^{-\sigma(t-s)}\abs{x(s)}\norm{B(s)}\dd s.
\]
\item[(b)] Let \(u(t)=e^{\sigma t}\abs{x(t)}\), \(v(t)=\norm{B(t)}\) and
\(c=Ke^{\sigma t_0}\abs{x_0}\). Show that the inequality of part (a) can be
rewritten as
\[
  u(t)\le c+\int_{t_0}^t v(s)u(s)\dd s.
\]
\item[(c)] From Gronwall's inequality we have
\[
  u(t)\le c\exp\left(\int_{t_0}^t v(s)\dd s\right).
\]
Use this to show that the zero solution of the IVP is asymptotically stable.
\end{enumerate}