A parsimonious mathematical model of pulse modulated regulation of non-basal testosterone secretion in the male is developed. The model is of third differential order, reflecting the three most significant hormones in the regulation loop, but is yet shown to be capable of sustaining periodic solutions with one or two pulses of gonadotropin-releasing hormone (GnRH) on each period. Lack of stable periodic solutions is otherwise a main shortcoming of existing low-order hormone regulation models. Existence and stability of periodic solutions are studied. The periodic mode with two GnRH pulses on the least period has not been described in medical literature but is found to explain experimental data well.

Library of Congress Cataloging-in-Publication Data Stability and oscillations of nonlinear pulse-modulated systems/ Arkadii Kh. Gelig and Alexander N. Churilov. editors. p. cm. Includes bibliographical references (p. -) and index.

Inst. Probl. Mech. Engg., Russian Ac. Sc. 61 Bolshoy V.O., St. Petersburg, 199178, RUSSIA alf@ccs.ipme.ru A b s t r a c t In this paper we give a survey of result,s related to the so called Feedback Kalman-Yakubovich Lemma (FKYL) giving necessary and sufficient solvability conditions for some class of bilinear matrix inequalities or conditions of feedback passivity of linear systems. Applications to adaptive and variable structure control systems are also discussed.

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