The Fenchel duality, $S$-procedure, and the Yakubovich–Kalman lemma

The role of the Fenchel duality in the theorem on the losslessness of the $S$-procedure and in Yakubovich–Kalman lemma is studied. The Fenchel duality theorem implying the well-known results on the losslessness of the $S$-procedure is formulated. A relation between the Yakubovich–Kalman lemma and a special extremal problem defined on a set of positive-demidefinite solutions of the generalized Lyapunov inclusion is derived. Every assertion of the lemma is shown to be necessary and sufficient for the value of the extremal problem to be bounded.