The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number , two n-vectors B, C and an n x n Hurwitz matrix A, if the pair is completely controllable, then a symmetric matrix P and a vector Q satisfying
exist if and only if
Moreover, the set is the unobservable subspace for the pair .
The lemma can be seen as a generalization of the Lyapunov equation in stability theory. It establishes a relation between a linear matrix inequality involving the state space constructs A, B, C and a condition in the frequency domain.