Kalman-Popov-Yakubovich lemma and the $S$-procedure: A historical essay

A history of two fundamental results of the mathematical system theory – the Kalman–Popov–Yakubovich lemma and the theorem of losslessness of the $S$-procedure – was presented. The studies directly concerned with these statements were reviewed. The recent publications using the theorem of losslessness of the $S$-procedure to derive the Kalman–Popov–Yakubovich lemma and its generalizations were analyzed.