## Homework 10 - Due: 04/16

### 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 \(x_1^2 + x_2^2+ x_3^2\).

### 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 *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 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: \[
\dot x = A(t) x + B(t) u
\]

### Problem 3

For a system

\[ \dot x = A(t) x(t) \]

we define the adjoint dynamics to be

\[ \dot p = -A^T(t) p(t) \]

Show that this definition

*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 \(\mathbb{R}^n\).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 \(M, N \in \mathbb{R}^{n \times m}\).

\[ \dot X = A_1(t)X(t) + X(t)A_2(t) \]

### 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 “*inverse*” to the original system by composing their transfer functions.

**Hint:** Use the Woodbury Matrix Identity.