Hw 5 ex 2

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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.

One can rewrite the Algebraic Riccati equation using {\tilde  {A}}, where you should note the changes of signs:

P{\tilde  {A}}+{\tilde  {A}}^{T}P+PBQ_{u}^{{-1}}B^{T}P+Qx=0

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.