Lyapunov Stability of the Generalized Stochastic Pantograph Equation

Abstract

The purpose of the paper is to study stability properties of the generalized stochastic pantograph equation, the main feature of which is the presence of unbounded delay functions. This makes the stability analysis rather different from the classical one. Our approach consists in linking different kinds of stochastic Lyapunov stability to specially chosen functional spaces. To prove stability, we check that the solutions of the equation belong to a suitable space of stochastic processes, instead of searching for an appropriate Lyapunov functional. This gives us possibilities to study moment stability, stability with probability 1, and many other stability properties in an efficient way. We show by examples how this approach works in practice, putting emphasis on delay-independent stability conditions for the generalized stochastic pantograph equation. The framework can be applied to any stochastic functional differential equation with finite dimensional initial conditions.

Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay

“…(ii) Asymptotic stability that is not exponential stability. is property is typical for equations with unbounded delays, for instance, for the stochastic pantograph equation [15]. e regularization method in its conventional setting has been used in this case, so that it is quite realistic to assume that combining it with the theory of M-matrices will not be difficult.…”

“…x(0) � b, (16) where (15). Note that f is uniquely defined by the stochastic process φ, "the prehistory function."…”

“…Its theoretical foundations are presented in the monograph [9], and since then numerous applications of the method have been investigated in a number of papers. Among the recent ones are the publications [10] (the Mackey-Glass equation), [11,12] (control theory), [13] (neural networks), [14] (hyperjerk systems), and [15] (the stochastic pantograph equation).…”

“…is led to new, verifiable stability conditions in the case of differential equations with variable, in particular, distributed delays. In general, the approach based on M-matrices gives better stability results as the technique utilizing the norm estimation (see, e.g., [15,17]). In the above publications, verifiable stability conditions were formulated in terms of the coefficients of the systems in question.…”

Stability of solutions of Ito linear difference equations systems with delay relative to the initial data

“…Идеи этого метода изложены в работе [4]. Этот метод на случай линейных стохастических функционально-дифференциальных уравнений распространен в работах [5][6][7]. В работе [4] этим методом исследованы вопросы устойчивости решений для систем линейных детерминированных разностных уравнений с последействием.…”