This image presents a problem about controllability and observability of a linear control system, represented by the following state-space equations:

$\frac{dx}{dt} = A x(t) + B u(t)$ $y(t) = C x(t)$

Where the matrices are given as:

$A = \begin{pmatrix} 0 & -1 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad C = \begin{pmatrix} -1 & 0 \end{pmatrix}$

The task is to study the controllability and observability of the system, and if the system is controllable, to construct one of the many control laws that drives the system from state $(0, x(0))$ to state $(1, 0)$.

Would you like to go through the solution for controllability and observability or explore the construction of a control law? Feel free to ask if you need more details!

Here are some related questions:

1. What are the conditions for controllability and observability in a state-space model?
2. How can we calculate the controllability matrix for a system?
3. How can we compute the observability matrix for a given system?
4. What is the role of the input matrix $B$ in controlling the system?
5. How do control laws help in state stabilization and system performance?

Tip: To assess controllability, we typically use the controllability matrix, and for observability, the observability matrix.