Math 6410 • ODE

\begin{enumerate}
\item[(a)] Let \(f(x)\in C^1(\R^n,\R^n)\) satisfy \(f(0)=0\). State the definitions:
the zero solution of
\[
  \begin{cases}
    \dfrac{dx}{dt}=f(x),\\
    x(0)=x_0,
  \end{cases}
\]
is stable in the sense of Liapunov and asymptotically stable.
\item[(b)] Suppose that all eigenvalues of the real \(n\times n\) matrix \(A\) have
negative real part \(\Reop\,\lambda_i<0\). State an estimate for solutions of
\[
  \begin{cases}
    \dfrac{dx}{dt}=Ax,\\
    x(0)=x_0,
  \end{cases}
\]
and use it to prove that the zero solution is asymptotically stable.
\item[(c)] Let \(A\) as in (b) and \(f\in C^1(\R^n,\R^n)\) be a smooth function such that
for some \(L<\infty\), \(\abs{f(x)}\le L\abs{x}^2\) for all \(x\). Consider the initial
value problem
\[
  \begin{cases}
    \dfrac{dx}{dt}=Ax+f(x),\\
    x(0)=x_0.
  \end{cases}
  \tag{3}
\]
Prove that the zero solution of (3) is stable in the sense of Liapunov. [Don't just
quote a theorem. 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 \(t\ge0\).]
\end{enumerate}