Construction of Lyapunov Functions for Second-Order Linear Stochastic Stationary Systems
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CONSTRUCTION OF LYAPUNOV FUNCTIONS FOR SECOND-ORDER LINEAR STOCHASTIC STATIONARY SYSTEMS M. M. Shumafov and V. B. Tlyachev
UDC 517.925.51, 519.216.73
Abstract. In this paper, we present a construction of Lyapunov functions for second-order linear stochastic systems with constant coefficients. Based on this construction, we state necessary and sufficient conditions of the mean-square exponential stability of two-dimensional linear stationary systems. We obtain analytical expressions for the bifurcation value of the intensity of white noise acting on the parameters of the system. As an example, we consider equations of elastic vibrations whose coefficients are perturbed by white noise. Keywords and phrases: stability, Lyapunov function, linear stochastic differential system, Gaussian white noise process. AMS Subject Classification: 93D05, 93E15
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Introduction
The method of Lyapunov functions or the direct Lyapunov method was originally proposed by A. M. Lyapunov (see [7]) for the analysis of the general problem of motion stability; later it was developed in works of numerous authors and became extremely general and powerful. The important role of Lyapunov functions in the theory of stability of deterministic systems is well known. The significance of this method is far from being exhausted by the possibility of establishing the fact of the stability or instability of the system under study. A successfully constructed Lyapunov function allows one to establish other properties of a system, for example, the boundedness of solutions, the dissipation property, the convergence of solutions, the existence or absence of periodic solutions, and so on. The works [4, 5] gave a powerful impulse to the development of the Lyapunov function method for the theory of stochastic differential equations. General theorems on basic qualitative properties, mainly, the stability property of stochastic systems, were proved by the method of stochastic Lyapunov functions. In these papers, the Lyapunov function method for deterministic systems was generalized to the stochastic case; it was the first step in constructing a qualitative theory of stochastic differential equations. We note that in all theorems proved in the works mentioned above, in order to establish one or another qualitative property of a system, the a priori existence of a Lyapunov-type function was supposed: if there exists a Lyapunov function with required properties, then the system under consideration possesses one or another property (stability, dissipativity, etc.). At the same time, the main problem, just as in the deterministic theory, is connected with the construction of a suitable Lyapunov function for the system under study in a given domain of the phase space. Until now, the problem of constructing an appropriate Lyapunov function for deterministic or stochastic systems has not been completely solved: there are no general methods for constructing Lyapunov functions. For deterministic systems, there exist several of methods for constructing Lyapunov functions (see, e.g.,
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