Kalman–Yakubovich–Popov lemma

The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number ${\displaystyle \gamma >0}$, two n-vectors B, C and an n x n Hurwitz matrix A, if the pair ${\displaystyle (A,B)}$ is completely controllable, then a symmetric matrix P and a vector Q satisfying

exist if and only if

Moreover, the set ${\displaystyle \{x:x^{T}Px=0\}}$ is the unobservable subspace for the pair ${\displaystyle (C,A)}$.

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