Asymptotic stability of solutions for a certain non-autonomous second-order stochastic delay differential equation

Abstract

In this paper, sufficient criteria that guarantee the existence of stochastic asymptotic stability of the zero solution of the nonautonomous second-order stochastic delay differential equation (1.1) were established with the aid of a suitable Lyapunov functional. Two examples are given in the last section to illustrate our main result.

Stability of High-Order Linear Itô Equations with Delays

“…Replacing the chain rule by Itô's formula leads to the same conclusion for the stochastic initial value problem (2), (5), (6). In particular, the exponential p-stability of Equation (2) follows from the exponential stability of system (13) for any 1 p ≤ < ∞ .…”

“…Stability of stochastic first-order differential equations with delays, as well as systems of equations, has been extensively studied (see [2] [4] [5] [6] [12] [13], [14] [15] [16] and the references therein). The main tool for studying the global stability is the Lyapunov functional method and its stochastic modifications (see e.g.…”

“…We define a solution of the initial value problem (2), (5), (6) to be a predictable stochastic process ( ),…”

“…In [4] explicit conditions for asymptotic stability of the second-order equation under additive white noise were obtained. In [6] boundedness and exponential stability conditions for second-order SDEs with a constant delay were examined. Other examples deal with the study of sensibility on stochastic perturbations of regenerative vibrations in milling process described by the second order linear differential equation with delays [7]; a planar inverted pendulum on a cart, operating under modelling uncertainties technically easier to obtain.…”

Asymptotic stability of solutions for a kind of third-order stochastic differential equations with delays

“…Meanwhile, the scarcity of works on stability and boundedness of solutions for third-order SDEs with or without delay were studied very rarely, interesting results are contained, for instance, in [1], [2], [3].…”