Math 6410 • ODE

Consider the linear non--autonomous first order system
\[
  \dot{x}=Ax+B(t)x,\qquad x\in\R^n
\]
with \(A\) non-singular and \(B(t)\) continuous for \(t\ge0\). Further, assume that
\begin{itemize}
\item the eigenvalues of \(A\) have non--positive real parts and those with zero real part
are non-degenerate
\item \(\displaystyle \int_0^\infty \lVert B(t)\rVert\dd t=c_1\) with \(c_1\) a positive
constant.
\end{itemize}
\begin{enumerate}
\item[(a)] Let \(\Phi(t)\) be the fundamental matrix of the equation \(\dot{x}=Ax\) with
\(\Phi(0)=I\). Derive the variation of constants formula
\[
  x(t)=\Phi(t)x(0)+\int_0^t \Phi(t-s)B(s)x(s)\dd s
\]
\item[(b)] Prove that the solution \(x(t)\) is bounded for all times \(t>0\).
[Hint: Use part (a) and Gronwall's lemma in the following form:
\(v(t)\le v_0+\int_0^t u(s)v(s)\dd s\) implies that
\(v(t)\le v_0\exp(\int_0^t u(s)\dd s)\) for \(t>0\)]. What does this imply about the
stability of the origin?
\end{enumerate}