1089the case, problem (15) is equivalent to problem (17). We further note that the inequality in (29) is equivalent to Both in the case of trace and log-determinant, the function f (X) is concave on the cone of positive-definite matrices. This implies that the optimal value of X; Z are X = Xopt , Z = Zopt , as claimed.
ACKNOWLEDGMENTThis note has benefited from interesting discussions and valuable input from several people, including R. Balakrishnan, S. Boyd, E. Féron, A. Kurzhanski, and R. Tempo. The authors particularly thank A. Nemirovski for his help regarding Section III-C. Useful comments from the reviewers and the Associate Editor are also gratefully acknowledged. Control, vol. 8, no. 4-5, pp. 377-400, 1998. [9] L. El Ghaoui and G. Calafiore, "Confidence ellipsoids for uncertain linear equations with structure," in 38th Conf. Decision Control, vol. 2, Phoenix, AZ, Dec. 1999, pp. 1922 , "Worst-case simulation of uncertain systems," Control, vol. 65, no. 5, pp. 847-866, 1996.
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On Kalman-Yakubovich-Popov Lemma for Stabilizable SystemsJoaquín Collado, Rogelio Lozano, and Rolf Johansson Abstract-The Kalman-Yakubovich-Popov (KYP) Lemma has been a cornerstone in System Theory and Network Analysis and Synthesis. It relates an analytic property of a square transfer matrix in the frequency domain to a set of algebraic equations involving parameters of a minimal realization in time domain. This note proves that the KYP lemma is also valid for realizations which are stabilizable and observable.
