Abstract
Ever since A. I. Lur'e introduced the absolute stability problem of nonlinear automatic control systems in his book in 1951, numerous researchers have proposed solutions and extensions of the problem, thus making it one of the most-researched problems in control theory. In my opinion, there are three crucial contributions that stand above all others. The frequency method introduced by V. M. Popov in 1960, the method of linear matrix inequalities proposed by V. A. Yakubovich in 1962, and Kalman's version of the positive real lemma formulated in 1963 via the concepts of controllability and observability. It has been the concept of matrix inequalities, however, that achieved special significance in a large number of areas in control theory and beyond. Supported by the recent advances in convex optimization, the matrix inequality methods of Yakubovich have dramatically changed the way we approach problems in control theory and applications. It is with great respect and pleasure that I offer to Professor Yakubovich my best wishes for his health and happiness on the occasion of his 80th birthday.
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