Abstract
The well-known Kalman--Yakubovich--Popov lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality; see [R. A. B. van der Geest and H. L. Trentelman, Systems Control Lett., 32 (1997), pp. 283--290; I. Masubuchi, Systems Control Lett., 55 (2006), pp. 158--164]. In this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of the so-called Lur'e equation [T. Reis, Linear Algebra Appl., 434 (2011), pp. 152--173] which mainly is a linear matrix inequality with a rank minimizing condition. While the results about the solvability of the algebraic Riccati equation [J. C. Willems, IEEE Trans. Automat. Control, 16 (1971), pp. 621--634] cannot be obtained from the Kalman--Yakubovich--Popov lemma we will see that with the Lur'e equation this is possible.
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