Math 6410 • ODE

Let \(A\) be an \(n\times n\) real matrix such that all solutions of \(\dot{x}=Ax\) are
bounded as \(t\to\infty\). Let \(B(t)\) be a continuous real \(n\times n\) matrix function
that satisfies
\[
  \int_0^\infty \norm{B(s)}\dd s<\infty.
\]
Consider the initial value problem
\begin{equation}
  \begin{cases}
    \dfrac{dx}{dt}=Ax+B(t)x\\
    x(0)=x_0
  \end{cases}
  \tag{1}
\end{equation}
\begin{enumerate}
\item[(a)] Define: the zero solution of (1) is \textit{stable} (also called uniformly
\textit{Liapunov Stable}).
\item[(b)] Prove that the zero solution of (1) is stable.
\end{enumerate}

[Don't just quote a theorem. Hint: Suppose \(g(t)\) and \(u(t)\) are nonnegative functions
and \(c_0\ge0\) is a constant that satisfies
\(u(t)\le c_0+\int_0^t g(s)u(s)\dd s\) 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 \(t\ge0\).]