40 Years of the Direct Matrix-Valued Lyapunov Function Method (Review)*
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International Applied Mechanics, Vol. 56, No. 3, May, 2020
40 YEARS OF THE DIRECT MATRIX-VALUED LYAPUNOV FUNCTION METHOD (REVIEW)*
À. À. Martynyuk
This review presents the main ideas and results of the development of the direct matrix-valued Lyapunov function method for studying the stability of continuous, singularly perturbed, and stochastic systems of equations of perturbed motion. Sufficient conditions of various types of stability for these systems of equations are expressed in terms of the properties of special matrices, which are obtained by constructing matrix-valued Lyapunov functions and evaluating their full derivatives. The equations of motion of an orbital astronomical observatory, the equations of motion of the generators of a large-scale power system, the three-component Lurie–Postnikov system, and others are applications of the results obtained. Keywords: matrix-valued Lyapunov function method, comparison principle, large-scale dynamic systems, stability, asymptotic stability, instability, polystability Introduction. The classical Lyapunov’s second method [9] in the qualitative theory of equations involves two stages: (i) constructing a Lyapunov function and (ii) establishing whether the total derivative of this function along the solutions of the perturbed equations of motion increase or decrease. As a result, based on Lyapunov’s general theorems, it is possible to qualitatively analyze the properties of the motion of the system of interest. The ideas of Lyapunov’s second method were developed in [26, 41–43] by using differential inequalities to analyze the global existence of solutions and the uniqueness and stability of the zero solution of a system of general nonlinear perturbed equations of motion. Lyapunov’s second method and differential inequalities were combined in [44] to establish stability conditions of the zero solution of the system of differential equations (1.1) in terms of the stability of the zero solution of the comparison equation. These studies included the basic ideas of the principle of comparison with a scalar Lyapunov function and were significantly developed by many researchers (see [45, 46, 81] and the references therein). The paper is organized as follows. Section 1 contains a preliminary analysis of the results obtained in developing the principle of comparison with scalar and vector Lyapunov functions. Section 2 discusses the necessity of introducing a matrix Lyapunov function and its typical features. Section 3 presents the results of development of Lyapunov’s second method for autonomous systems based on an auxiliary matrix function. The basic theorems of dynamic properties are formulated based on “maximum functions” defined over a set of elements of a matrix Lyapunov function.
S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterov St., Kyiv, Ukraine 03057; e-mail: [email protected]. Translated from Prikladnaya Mekhanika, Vol. 56, No. 3, pp. 3–75, May–June 2020. Original article submitted May 6, 2019.
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