\subsection{Due: 02/15}

\subsubsection{Problem 1}

In class last week, we discussed changes to the LTV system
\(\dot x = A(t) x(t)\) under a time varying coordinate change
\(x(t) = P(t) \bar{x}(t)\) and derived a form for the \(\bar{A}(t)\)
matrix in the equivalent representation:
\(\dot {\bar{x}} = \bar{A}(t) \bar{x}(t)\) under the assumption that
\(P(t)\) is invertible for all \(t\).

\[
 \bar{A}(t) = \dot{P}^{-1}(t) P(t) + P^{-1}(t)A(t)P(t)
\]

For this problem

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  Extend the coordinate change to the full system: \[
   \dot x = A(t) x(t) + B(t) u (t)
   \]
\item
  Show that if \(P(t)\) is a fundamental matrix for the system \[
   \dot{P}(t) = A(t) P(t), \qquad P(t_0) = C \in \mathbb{R}^{n\times n}
   \] where \(C\) is invertible then \[
   \dot{\bar{x}} (t) = P^{-1}B(t) u(t) 
   \]
\item
  Using this result re-derive the variation of constants formula \[
   x(t) = \phi(t, t_0)x(t_0) + \int \limits _{t_0} ^{t} \phi (t, s) B(s) u(s) ds 
   \] where \(\phi (t, s) = P(t)P^{-1}(s)\).
\end{enumerate}

\subsubsection{Problem 2}

Prove that the Euclidean norm \[|x|:=\sqrt{\langle x,x\rangle}
=\sqrt{x_1^2+\dots+x_n^2}\] satisfies the triangle inequality.

\textbf{Hint:} Use the Cauchy-Schwarz inequality
\(|\langle x,y\rangle|^2\le \langle x,x\rangle\cdot \langle y,y\rangle\).

\subsubsection{Problem 3}

Let \(A\) be a symmetric real-valued square matrix.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that if \(\lambda+i\mu\) is an eigenvalue of \(A\) and \(z=x+iy\)
  is a corresponding eigenvector, then \(\mu=0\) and \(x\) is an
  eigenvector (you can assume \(x\ne 0\) for simplicity). In other
  words, eigenvalues of symmetric matrices are always real and
  eigenvectors can always be chosen to be real.

  \textbf{Hint}: Show that \(\overline z^T\!Az\) is real.
\item
  Show that eigenvectors of \(A\) corresponding to distinct eigenvalues
  are orthogonal.
\end{enumerate}

\subsubsection{Problem 4}

Let \(X\) and \(Y\) be linear vector spaces over \(\mathbb R\) equipped
with inner products \(\langle\cdot,\cdot\rangle_X\) and
\(\langle\cdot,\cdot\rangle_Y\), respectively. Further, let \(L:X\to Y\)
be a linear operator.

We define the \textbf{adjoint} of \(L\) to be a linear operator
\(L^*:Y\to X\) with the property that \[
\langle y,Lx\rangle_Y=\langle L^*y,x\rangle_X\qquad \forall\, x\in X,\ y\in Y
\] Assume that the map \(LL^*:Y\to Y\) is invertible. Then the equation
\(Lx=y_0\) has a solution \[
x_0=L^*(LL^*)^{-1}y_0
\] for each \(y_0\in Y\).

Prove that if \(x_1\) is any other solution of \(Lx=y_0\), then
\(\langle x_1,x_1\rangle\ge \langle x_0,x_0\rangle\).

\textbf{Hint:} Let \(y_1:=(LL^*)^{-1}y_0\). Using the definition of
adjoint, show that \(\langle y_1,Lx_0\rangle=\langle x_0,x_0\rangle\)
and also that \(\langle x_0,x_1\rangle=\langle y_1,Lx_0\rangle\)
Complete the proof by using the fact that
\(\langle x_1-x_0,x_1-x_0\rangle\ge 0\).

\subsubsection{Problem 5}

Using the stability definitions given in class, determine if the systems
below are stable, asymptotically stable, globally asymptotically stable,
or neither. The first two systems are in \(\mathbb{R}^2\), the last is
in \(\mathbb{R}\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  \(\dot x_1 = 0\) and \(\dot x_2 = -x_2\)
\item
  \(\dot x_1 = -x_2\) and \(\dot x_2 = 0\)
\item
  \(\dot x = 0\) if \(|x|>1\) and \(\dot x = -x\) otherwise
\end{enumerate}

Justify your answers using \emph{only} the definitions of stability (not
eigenvalues or Lyapunov's method).

\subsubsection{Problem 6}

First, some definitions.

\begin{itemize}
\item
  Given a linear operator \(A:X\to X\), a subspace \(Y\subset X\) is
  called \(A\)-\emph{invariant} if \(Ay\in Y\) \(\forall y\in Y\).
\item
  For a linear system \(\dot x=Ax\) on \(X=\mathbb{R}^n\), this means
  that \(x_0\in Y\) implies \(x(t)\in Y\) \(\forall t\) (reason:
  \(x(t)=e^{At}x_0=(I+A+A^2/2+\dots)x_0\)).
\item
  If \(v\) is an eigenvector of \(A\) with a real eigenvalue, then
  \(\operatorname{span}\{v\}\) is a 1-dimensional invariant subspace.
  For a \(k\times k\) Jordan block, the eigenvector \(v_1\) and the
  generalized eigenvectors \(v_2,\dots, v_k\) together span an invariant
  subspace.
\item
  The case of a pair of complex eigenvalues was discussed in
  \href{/secure/homework/hw02.html\#problem-5}{Problem Set 2, Problem 3}
\item
  The sum of all invariant subspaces corresponding to eigenvalues with
  \(\text{Re}(\lambda)<0\) is called the \emph{stable invariant
  subspace}; the corresponding object for \(\text{Re}(\lambda)\ge 0\) is
  the \emph{unstable invariant subspace}; together these two subspaces
  span \(\mathbb{R}^n\).
\end{itemize}

Consider the LTI system \(\dot x =Ax\) where \[
A=
  \begin{pmatrix}
    -1  & 0 & 0 \\
    0 & 2 & 1\\
    0 & -1 & 2
  \end{pmatrix}
\] Identify the stable and unstable invariant subspaces by giving a
\emph{real} basis for each of them.
