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The Kalman-Yakubovich-Popov (KYP) Lemma, giving the frequency-domain criteria for solvability of LQR problems and feasibility of related LMIs and Luru0027e-Riccati equations, is undoubtedly one of the cornerstone results in modern control theory. Numerous applications of the KYP lemma in nonlinear and robust control motivated its extension to systems with distributed parameters. Unlike the finite-dimensional case, where the KYP lemma admits elegant proofs by means of algebraic techniques and methods from convex analysis, a prerequisite for its infinite-dimensional extensions is feasibility of the corresponding LQR problem, resulting on assumptions of exponential or L 2 -stabilizability. This property is restrictive and not easily verifiable for general infinite-dimensional systems. Being quite natural in the problems of absolute stability, it seems unnecessary in other applications of the KYP lemma which require only solvability of the operator inequality, but not positivity of its solution, such as e.g. instability or dichotomy criteria. In the present paper we show, that in the “strict” or coercive case the KYP lemma retains its validity, replacing stabilizability assumption with some technical assumption which holds, for instance, whenever the linear system is exponentially dichotomic.

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A new extension of the infinite-dimensional KYP lemma in the coercive case∗∗The paper was partially supported by RFBR, grants 13-0801014 and 14-08-01015, and St. Petersburg State University, grant 6.38.230.2015. Theorem 10 in Section 4 is supported solely by Russian Scientific Foundation (RSF), grant 14-29-00142.