Math 6410 • ODE

Let \(A\) be an \(n\times n\) real matrix whose eigenvalues \(\lambda_i\) satisfy
\(\Re\lambda_i<0\) for all \(i=1,\ldots,n\). Let \(B(t)\) be a smooth real \(n\times n\)
matrix function and \(\beta>0\) such that \(\lvert B(t)\rvert\le e^{-\beta t}\) for all
\(t\ge0\). Consider the initial value problem for \(x_0\in\R^n\),
\begin{equation}
  \begin{aligned}
    \frac{dx}{dt} &= [A+B(t)]x,\\
    x(0) &= x_0.
  \end{aligned}
  \tag{2}
\end{equation}
\begin{enumerate}
\item[(a)] Show that the solution to (2) exists for all time and
\(\lim_{t\to\infty}x(t)=0\).

[Hint: 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. Define what it means for
\(z\) to be \textit{Asymptotically Stable}. Show that \(z\) is asymptotically stable for
this equation.
\end{enumerate}