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\title{Differential Equations}
\author{Summer Semester 2024, Exercise 3}
\date{Due Monday, May 6, 2024}
\maketitle

\begin{enumerate}

\item Let $A \in \R^{n \times n}$ such that $\Re \lambda \leq \alpha$
for every eigenvalue $\lambda$ of $A$.  Let $k$ be the dimension of
the biggest Jordan block in the Jordan canonical form of $A$.  Show
that there exists a constant $C$ such that
\[
  \lVert \e^{At} \rVert
  \leq C \, \e^{\alpha t} \,
  \biggl(
    1 + t + \dots + \frac{t^{k-1}}{(k-1)!} 
  \biggr) \,.
\]
\emph{Hint:} First prove the result for a single Jordan block (using
the solution formula from class; also see Teschl, equation 3.18).
Then argue for general $A$.

\emph{Note:} This bound immediately implies the upper bound
\[
  \lVert \e^{At} \rVert
  \leq \tilde C \, \e^{\alpha t} \, ( 1 + t^{k-1})
\]
for some other constant $\tilde C$.  (Why? -- For discussion only.)

\item Consider the damped harmonic oscillator in the form
\[
  \ddot q + \dot q + q = 0 \,.
\]
\begin{enumerate}
\item Show that this equation can be written in the form
\[
  \dot x = A x
\]
with
\[
  A =
  \begin{pmatrix}
  0 & 1 \\ -1 & -1
  \end{pmatrix} \,.
\]

\item Show that $A$ satisfies the assumption of Problem~1 above.  What
is $\alpha$, what is $k$?
\item Show that $A$ is \emph{not} strictly negative definite.  (A
matrix $A \in \R^{n \times n}$ is strictly negative definite if
$x^TAx<0$ for all $x \in \R^n$.)
\item Show that rate of change of the energy $E = x^Tx$ is not
strictly negative at all times.
\item Give a physical interpretation of this fact.  Moreover, argue
that it is nonetheless consistent with the conclusion of Problem~2.
\end{enumerate}

\item Show that if all eigenvalues of a matrix $A$ have negative real
part, but $A$ is not diagonalizable (i.e., its Jordan canonical form
has at least one block of dimension $k>1$), then the corresponding
differential equation
\[
  \dot x = Ax
\]
has ``transient growth'', i.e., there are initial conditions such that
$E=x^Tx$ is growing on some interval of time, even through (by the
result from Problem~1 above) $E(t) \to 0$ as $t \to \infty$.


\end{enumerate}
\end{document}
