Noise expresses exponential decay for globally exponentially stable nonlinear time delay systems

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Abstract

In this paper, we show that noise can make a given nonlinear time delay system which is globally exponentially stable become a new stochastic one whose solution to exponential decay faster than the former. By comparing the upper bounds of noise intensity and coefficients of global exponential stability, we could deduce that noise is able to further express exponential decay for nonlinear time delay system without losing global stability. The upper bounds of noise intensity are characterized by solving transcendental equations containing adjustable parameters. In addition, a numerical example is provided to illustrate the theoretical result.

Introduction

The nonlinear time delay systems in many branches of science and industry do not only depend on the present state but also the past ones. In recent decades, the nonlinear delay systems have been developed and applied extensively in many fields such as physics, engineering, mechanics, pattern recognition, associative memories, aerospace and robotics and control [9], [10], [15], [16], [20], [22], [25], [32], [39], [34]. On the other hand, these systems may often go through abrupt changes in their structure and parameters have been used to model these abrupt changes.

It is well known that noise can be used to stabilize a given unstable system or to make a system even more stable when it is already stable. There is an extensive literature concerned with the stabilization by noise, e.g., [1], [2], [23], [26], [27], [28], [33], [41], [42], [44], [45], [46] and the references therein. The pioneering work in this area is given due to Hasminskii [10], who stabilized an unstable system by using two white noise sources. Several years ago, Mao et al. [24] showed another important fact that the environmental noise can suppress explosions in a finite time in population dynamics. Recently, Deng et al. [8] revealed that the noise can suppress or expresses exponential growth under the linear growth condition. In [11], [12], Hu et al. developed the theory in [8] to cope with much more general systems. Without the linear growth condition, Wu and Hu [36] further considered the problems of stochastic suppression and stabilization of nonlinear differential systems. Liu and Shen [21] revealed that the single noise can also make almost every path of the solution of corresponding stochastically perturbed system grow at most polynomially.

Noises can lead to instability and they can destabilize stable nonlinear delay systems if it exceeds their limits. The instability depends on the intensity of noise. For a stable nonlinear delay system, if the intensity of noise is low, the stochastic nonlinear delay system may become even more stable. Therefore, it is interesting to determine how much random disturbances which can further express exponential decay for a stable nonlinear delay system. Although the various stability properties of nonlinear delay systems by noise have been extensively investigated using the Lyapunov stability theory [19], [38], [29], [30], [40], [43], the linear matrix inequality methods [3], [4], [5], [6], [7], [13], [14], [18], [35], [37] and the comparison principle theory [17], it is rarely analyzed directly by estimating the upper bounds of noise level from the coefficients of globally exponential stability condition that noise makes the nonlinear delay system even more stable when it is already stable.

Motivated by the above discussion, our aim in this paper is to quantify the parameter uncertainty level for stable nonlinear delay systems. Different from the traditional Lyapunov stability theory and the matrix norm theory, we investigate that noise expresses exponential stability of nonlinear delay systems directly from the coefficients of the nonlinear delay systems should be satisfied the globally exponential stability condition. In this paper, we could deduce that noise is able to further express exponential decay for nonlinear time delay system without losing global stability. By comparing the upper bounds of noise intensity and coefficients of global exponential stability. The upper bounds of noise intensity are characterized by solving transcendental equations containing adjustable parameters.

Section snippets

Problem formulation

Throughout this paper, unless otherwise specified, Rn and Rn×m denote, respectively, the n-dimensional Euclidean space and the set of n×m real matrices. Let (Ω,F,{Ft}t≥0,P) be a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e. the filtration contains all P-null sets and is right continuous). ω(t)=(ω1(t),ω2(t),…,ωn(t))T,t≥0 is an n-dimensional standard Brownian motion defined on the probability space. If A is a matrix, its operator norm is denoted by ∥A∥=

Main results

Now, the question is, given a globally exponentially stable nonlinear time delay system. Under the noisy condition, how much the intensity the nonlinear delay system can bear? we consider the noise-induced nonlinear delay system described by the n-dimensional stochastic functional differential equation of Itô type{dy(t)=f(y(t),y(t−τ(t)),t)dt+g(y(t),t)dω(t),t>t0,y(t)=ψ(t−t0),t0−τ¯≤t≤t0,where the notations of (4) are the same as in Section 2, f satisfies Assumption 1, g:Rn×Rn→Rn×n is a Borel

Illustrative example

Example 1

Consider a two-dimensional nonlinear delay system{x1̇(t)=−x1(t)−f(x1(t))+f(x2(t))−f(x1(t−τ(t)))+f(x2(t−τ(t))),x2̇(t)=−x2(t)+f(x1(t))−f(x2(t))+f(x1(t−τ(t)))−f(x2(t−τ(t))),where f(xj)=(|xj+1|−|xj−1|)/2,τ(t)=0.00025(sint+1),x(0)=[0.4,−0.2]T. Hence, according to Theorem 2 in [31], nonlinear delay system (30) is globally exponentially stable with α=0.4, β=0.0699. In the presence of noise, the delay system becomes a stochastic delay system:{y1(t)dt=[−y1(t)−f(y1(t))+f(y2(t))−f(y1(t−τ(t)))+f(y2(t−τ(t)))

Conclusion

In this paper, we discuss whether or not the intensity of noise which makes the globally exponentially stable nonlinear time delay system even more stable. Employing the coefficients of global exponential stability, we deduce that noise is able to further express exponential decay for the nonlinear delay system without losing global stability. The upper bounds of noise can be estimated by solving transcendental equations containing adjustable parameters. It can result conservative result as we

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities of 2015QN55.

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