Math 6410 • ODE

Consider the following linear equation for \(x\in\mathbb{R}^n\):
\[
\dot{x}=Ax+B(t)x,\quad x(t_0)=x_0.
\]
where \(A\), \(B(t)\) are \(n\times n\) matrices. Suppose that all eigenvalues
\(\lambda_j\), \(j=1,\ldots,n\), of the matrix \(A\) satisfy
\(\operatorname{Re}(\lambda_j)<0\), and let \(B(t)\) be continuous for
\(0\le t<\infty\) with
\[
\int_0^\infty \|B(t)\|\,dt<\infty.
\]
\begin{enumerate}

\item
Using the variation of constants formula show that there exist constants
\(K,\sigma>0\) such that
\[
|x(t)| \le K e^{-\sigma(t-t_0)}|x_0|
  +K\int_{t_0}^t e^{-\sigma(t-s)}|x(s)|\|B(s)\|\,ds.
\]


\item
Let \(u(t)=e^{\sigma t}|x(t)|\), \(v(t)=\|B(t)\|\) and
\(c=Ke^{\sigma t_0}|x_0|\). Show that the inequality of part (a) can be
rewritten as
\[
u(t)\le c+\int_{t_0}^t v(s)u(s)\,ds.
\]


\item
From Gronwall's inequality we have
\[
u(t)\le c\exp\left(\int_{t_0}^t v(s)\,ds\right).
\]
Use this to show that the zero solution of the initial value problem is asymptotically
stable.

\end{enumerate}