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.
Multivariable Kalman–Yakubovich–Popov lemma
Given with for all and controllable, the following are equivalent:
- for all
- there exists a matrix such that and
The corresponding equivalence for strict inequalities holds even if is not controllable. 
- Kalman, Rudolf E. (1963). "Lyapunov functions for the problem of Lur'e in automatic control" (PDF). Proceedings of the National Academy of Sciences. 49 (2): 201–205. doi:10.1073/pnas.49.2.201.
- "Anders Rantzer" (1996). "On the Kalman–Yakubovich–Popov lemma". Systems & Control Letters. 28 (1): 7–10. doi:10.1016/0167-6911(95)00063-1.