Math 6410 • ODE

Let \(A\) be an \(n\times n\) real matrix such that all eigenvalues \(\lambda_i\) have
negative real parts \(\Re\lambda_i<0\). Let \(f\in C^1(\R^n,\R^n)\) be a smooth function
such that \(f\) and its derivative \(df\) vanish at zero (\(f(0)=0\) and \(d_0f=0\)).
Consider the initial value problem
\begin{equation}
  \begin{cases}
    \dfrac{dx}{dt}=Ax+f(x),\\
    x(0)=x_0
  \end{cases}
  \tag{1}
\end{equation}
\begin{enumerate}
\item[(a)] Define: the zero solution of (1) is asymptotically stable.
\item[(b)] Prove that the zero solution of (1) is asymptotically 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 \(t\ge0\).]