The Kalman–Yakubovich–Popov Lemma (also called the Yakubovich–Kalman–Popov Lemma) is considered to be one of the cornerstones of Control and Systems Theory due to its applications in absolute stability, hyperstability, dissipativity, passivity, optimal control, adaptive control, stochastic control, and filtering. Despite its broad field of applications, the lemma has been motivated by a very specific problem which is called the absolute stability Lur’e problem [1, 2], and Lur’e’s work in [3] is often quoted as +being the first time the so-called KYP Lemma equations have been introduced. The first results on the Kalman–Yakubovich–Popov Lemma are due to Yakubovich [4, 5] . The proof of Kalman [6] was based on factorization of polynomials, which were very popular among electrical engineers. They later became the starting point for new developments. Using general factorization of matrix polynomials, Popov [7, 8] obtained the lemma in the multivariable case. In the following years, the lemma was further extended to the infinite-dimensional case (Yakubovich [9], Brusin [10], Likhtarnikov and Yakubovich [11]) and discrete-time case (Szego and Kalman [12]).

Call for Papers Special Issue: ‘Dissipativity of Dynamical Systems: Application in Control’ Dedicated to VasHe Mihai Popov

Kalman-Yakubovich-Popov Lemma

This paper investigates the existence and potential uniqueness of equilibrium points and some stability, instability and oscillatory properties of a discrete nonlinear epidemic model, which generalises previous Stevic’s model, whose solutions possess memory from a finite chain of preceding samples. An application example is provided where the proposed model is ‘ad hoc’ adapted to a class of SIS models widely used in epidemiology.

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Call for Papers Special Issue: ‘Dissipativity of Dynamical Systems: Application in Control’ Dedicated to VasHe Mihai Popov

Citations

Kalman-Yakubovich-Popov Lemma

Some equilibrium, stability, instability and oscillatory results for an extended discrete epidemic model with evolution memory

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