Extended Kalman-Yakubovich-Popov lemma in a Hilbert space and Fenchel duality.

The Kalman-Yakubovich-Popov (KYP) lemma is extended with new conditions that are equivalent to solvability of the Lur'e equation or the corresponding linear operator inequality. The relation established between the KYP lemma and an extremum problem on the set of positive semi-definite solutions of the generalized Lyapunov inclusion. It is proved that the statements of the KYP lemma are necessary and sufficient conditions for value to be bounded in this problem. The approach is based on the special Fenchel duality theorem and presents the new proof of the KYP lemma as well. The linear-quadratic optimization problem for a behavioral system in a Hilbert space is considered to illustrate the application of the new statements that are added to the KYP lemma.