Problem 8
(Solution to an LTV system). Consider the homogeneous linear time-varying system
$$\dot{x} = A(t)x,$$
$$x(0) = x_0$$
with state transition matrix $\Phi(t, \tau)$. Consider also the non-homogeneous system
$$\dot{z} = A(t)z + x(t),$$
$$z(0) = z_0$$
whose input $x(t)$ is the state of the homogeneous system.
(a) Compute $x(t)$ and $z(t)$ as a function of $x_0$, $z_0$, and $\Phi$. No integrals should appear in your
answer.
(b) For a given time $T> 0$, how should $x_0$ and $z_0$ be related to have $z(T) = 0$?
