\subsection{Homework 10 - Due: 04/16}
\subsubsection{Problem 1}
Find the solution of the vector equation \[
\begin{bmatrix}
3 &2 &1 \\
5 &2 &1
\end{bmatrix} \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}
\]
which minimizes \(x_1^2 + x_2^2+ x_3^2\).
\subsubsection{Problem 2}
In class, we relied on a theorem from Luenberger's OVSM to establish
that the control law \[
u(t) = B^T(t) \phi^T (t_0, t) \eta, \qquad \eta = W^{-1}(t_0, t_1) \left( \phi(t_0, t_1)x_1 - x_0 \right)
\] (appropriately specialized for the LTI case) sending \(x_0\) at time
\(t_0\) to \(x_1\) at time \(t_1\) minimizes the \emph{energy}: \[
J(u) = \int \limits _{t_0} ^{t_1} u^2(t) dt
\] The above functional can also be interpreted as the squared \(L_2\)
norm of a function \(u(t)\). Use the result of
\href{hw04.html\#problem-4}{Problem 4 from Homework 4} to provide a
proof establishing that the same control law minimizes the \(L_2\) norm
for LTV systems as well: \[
\dot x = A(t) x + B(t) u
\]
\subsubsection{Problem 3}
For a system
\[
\dot x = A(t) x(t)
\]
we define the adjoint dynamics to be
\[
\dot p = -A^T(t) p(t)
\]
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
Show that this definition \emph{preserves} (i.e.~holds constant) the
inner product between solutions \(x(t)\) and \(p(t)\). Here the inner
product is the standard inner product in \(\mathbb{R}^n\).
\item
Derive the adjoint dynamics \(\dot P\) for the matrix differential
equation below which preserves the inner product
\(\langle M, N \rangle = \operatorname{tr}(M^TN) = \operatorname{tr}(N^TM)\)
between matrices \(M, N \in \mathbb{R}^{n \times m}\).
\end{enumerate}
\[
\dot X = A_1(t)X(t) + X(t)A_2(t)
\]
\subsubsection{Problem 4}
Consider a dynamical system \[
\dot x = Ax + bu, \qquad y = cx + u
\] Show that the sytem \[
\dot x = \left(A - b c \right) x + by, \qquad u = -cx + y
\] is an ``\emph{inverse}'' to the original system by composing their
transfer functions.
\textbf{Hint:} Use the
\href{https://en.m.wikipedia.org/wiki/Woodbury_matrix_identity}{Woodbury
Matrix Identity}.