\subsection{Due: 02/08}

\subsubsection{Problem 1}

Consider the following matrices:

\[
A_1 = \begin{bmatrix} 1/2 & -1/2 \\ 1/2 & -1/2 \end{bmatrix}, \qquad  
A_2=\begin{bmatrix}0&1\\-1&0\end{bmatrix},\qquad
A_3=\begin{bmatrix}1&0&0\\0&-2&0\\-3&0&-2\end{bmatrix}.
\]

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  Compute \(e^{A_i t}\) for \(i=1,2,3\). Note that \(A_2\) is a special
  case of a matrix whose exponential was computed in class.
\item
  For \(i=1,2,3\), write down the solution of \(\dot x=A_i x\) for a
  general initial condition.
\item
  In each case, determine whether the solutions of \(\dot x=A_i x\)
  decay to 0, stay bounded, or go to \(\infty\) (for various choices of
  initial conditions).
\item
  Try to state the general rule which can be used to determine, by
  looking at the eigenstructure of \(A\), whether the solutions of
  \(\dot x=A x\) decay to 0, stay bounded, or go to \(\infty\).
\end{enumerate}

\subsubsection{Problem 2}

The pictures in this
\href{http://cstd.ece.illinois.edu/phase_plots.html}{demo page} show
possible trajectories of a linear system \(\dot x= Ax\) in the plane,
when the eigenvalues \(\lambda_1\) and \(\lambda_2\) of \(A\) are in one
of the following \textbf{six} configurations:

\begin{longtable}[]{@{}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.0690}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.4253}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.0690}}
  >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.4368}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
\#
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Configuration
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\#
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Configuration
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
(a) & \(\lambda_1<\lambda_2<0\) & (b) & \(\lambda_1<0<\lambda_2\) \\
(c) & \(0<\lambda_1<\lambda_2\) & (d) &
\(\lambda_1,\lambda_2=a\pm ib, \; a<0\) \\
(e) & \(\lambda_1,\lambda_2=a\pm ib, \; a=0\) & (f) &
\(\lambda_1,\lambda_2=a, \; a>0\) \\
\end{longtable}

Match each picture with the corresponding eigenvalue distribution.
Justify your answers using your knowledge of the solution of
\(\dot x = Ax\). Plotting in MATLAB or use of the demo applet cannot be
used as a justification.

\begin{figure}
\centering
\includegraphics[width=0.65\textwidth,height=\textheight]{./figures/hw3_2x2systems.gif}
\caption{Animations courtesy of
\href{https://personal.math.ubc.ca/~israel/}{Prof.~R.B. Israel}}
\end{figure}

\subsubsection{Problem 3}

This exercise illustrates the phenomeom known as resonance. Consider the
system

\begin{align*}
\begin{bmatrix}
   \dot x_1\\
    \dot  x_2
  \end{bmatrix} &=
  \begin{bmatrix}
    0 & \omega \\
    -\omega & 0
  \end{bmatrix} \begin{bmatrix}
    x_1\\
    x_2
  \end{bmatrix} +\begin{bmatrix}
    0\\
    1
  \end{bmatrix}u,\qquad
  y=x_2
\end{align*}

where \(\omega>0\), \(u\) is the control input, and \(y\) is the output.
Let the input be \(u(t)=\cos\nu t\), \(\nu>0\). Using the
variation-of-constants formula, compute the response \(y(t)\) to this
input from the zero initial condition \(x(0)=0\), considering separately
two cases: \(\nu\ne\omega\) and \(\nu=\omega\). In each case, determine
whether this response is decaying to 0, bounded, or unbounded.

\subsubsection{Problem 4}

Compute the state transition matrix \(\Phi(t,t_0)\) of \(\dot x=A(t)x\)
with \[A(t) = \begin{bmatrix}-1+\cos t&0 \\0&-2+\cos t \end{bmatrix}\]

\subsubsection{Problem 5}

Consider the LTV system \(\dot x=A(t)x\), where \(A(t)\) is
\emph{periodic} with period \(T\), i.e., \(A(t+T)=A(t)\). Let
\(\Phi(t,t_0)\) denote the corresponding state transition matrix. The
goal of this exercise is to show that we can simplify the system and
characterize its transition matrix with the help of a suitable
(time-varying) coordinate transformation.

Being a nonsingular matrix, \(\Phi(T,0)\) can be written as an
exponential: \(\Phi(T,0)=e^{RT}\), where \(R\) is some (possibly
complex-valued) matrix. Define also:

\[P(t):=\Phi(t,0)e^{-Rt}\]

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  Using the properties of state transition matrices from class, show
  that \(\Phi(t,t_0)=P(t)e^{R(t-t_0)}P^{-1}(t_0).\)
\item
  Deduce from a) that \(\bar x(t):=P^{-1}(t)x(t)\) satisfies the
  time-invariant differential equation \(\dot{\bar x}=R\bar x\).
\item
  Prove that \(P(t)\) is periodic with period \(T\).
\item
  Consider the LTV system from Problem 4 . Find new coordinates
  \(\bar x(t)\) in which, according to part b), this system should
  become time-invariant. Confirm this fact by differentiating
  \(\bar x(t)\).
\end{enumerate}

\subsubsection{Problem 6}

Prove the variation-of-constants formula for linear time-varying control
systems stated in class: \[
x(t) = \phi(t,t_0)x_0 + \int \limits _{t_0} ^{t} \phi \left(t, s \right) B(s) u(s) ds
\] satisfies the LTV controlled systems' differential equation.

\textbf{Hint:} Differentiate both sides. See also Class Notes, end of
Section 3.7.
