Hw 5 ex 2

Hint on how to solve ex 2: assume that the system is observable, and try an argument by contradiction. If the controller makes the system unstable, then the corresponding matrix $\tilde{A}=A-BK$ must have an eigenvalue with positive real part, to which corresponds a certain eigenvector $v$. Writing down the algebraic Riccati equation with $\tilde{A}$, and pre-post multiplying by the unstable eigenvector (as if you were evaluating a quadratic form), you will see that the only case in which the corresponding form can be zero is only if P=0 and $v^{*}{Q}_{v}v$ is zero. Which contradicts the initial assumption.

--Elisa