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
\(g(x,t)\in C^\infty(\R^n\times\R,\R^n)\) such that
\(\lvert g(x,t)\rvert\le \lvert x\rvert^2\) for all \(x\in\R^n\). Consider the ODE
\[
  y'=Ay+g(y,t).
\]
\begin{enumerate}
\item[(a)] Show that if the initial value \(\lVert x(0)\rVert\) is small enough, then
the solution exists for all time and \(\lim_{t\to\infty}x(t)=0\). [Hint: Gronwall's
Inequality for nonnegative functions \(u(t)\) and \(v(t)\) that satisfy
\(v(t)\le c_0+\int_0^t u(s)v(s)\dd s\) is
\(v(t)\le c_0\exp\left(\int_0^t u(s)\dd s\right)\).]
\item[(b)] Let \(z(t)=0\) for all \(t\) be the zero solution. Define what it means for \(z\)
to be \textit{Stable} (often called \textit{Liapunov Stable}.) Discuss the stability of
\(z\) for this equation.
\end{enumerate}