Research paper
Quantitative bounds for general Razumikhin-type functional differential inequalities with applications

https://doi.org/10.1016/j.cnsns.2020.105253Get rights and content

Highlights

  • Explicitly propose the concept of Razumikhin-type functional differential inequalities (RFDIs) for the first time.

  • Propose a more relaxed condition for studying quantitative bounds for the RFDIs.

  • Establish general and uniform bounds for general Razumikhin-type differential inequalities.

  • Present new more direct approach to deduce some basic Razumikhin-type stability results than the classical ones.

Abstract

This paper proposes the concept of Razumikhin-type functional differential inequalities and points out that certain quantitative properties can be established for the Razumikhin-type functional differential inequalities. By virtue of certain auxiliary functions, some fundamental results on the quantitative bounds for the Razumikhin-type functional differential inequalities are systemically established in the paper, and these bounds are applied to deduce the basic Razumikhin-type stability theorems, including those for Ito^ stochastic functional differential equations. Two examples are given to illustrate the application of the established quantitative properties and to verify the effectiveness of our approach.

Introduction

The well-known Razumikhin technique provides us an approach to overcome the difficulties brought about by time delays so as to establish stability theorems or criteria for functional differential equations. This technique was initially proposed by Razumikhin [1], [2] to study the stability of deterministic functional differential equations. Later, this technique was further investigated by Hale [3], Hou and Qian [4], [5], Hou and Gao [6], Teel [7], Sun et al. [23] and extended to some other models, such as stochastic neutral models [8], switching models [9], models driven by Levy noise [16] and impulsive models [22], [31], [34]. Especially in applications, several generalized problems have been considered, such as H control [25] and adaptive feedback control [32], stabilization [27], [30], [33]. The Razumikhin technique has become very popular in recent years since it is extensively applied in the fields of applied mathematics and control engineering. The corresponding results are generally referred to as stability theorems of the Razumikhin type.

For the functional differential equation{x˙(t)=f(t,xt),tt0,xt0(θ)=ϕ0(θ),θIτ=[τ,0], where f(t,ϕ)C(R+×C(Iτ;Rn);Rn) is a completely continuous functional with f(t,0)=0Rn for all t ≥ t0. Assume that for every initial condition ϕ0C(Iτ;Rn), there exists a unique global solution to the Eq. (1), which is denoted by x(t)=x(t;t0,ϕ0). So, under the assumption f(t,0)=0 for all t ≥ t0, the Eq. (1) has the solution x(t) ≡ 0 corresponding to the initial condition ϕ0(θ)=0,θIτ. This solution is called the trivial solution.

For the above Eq. (1), recall the Razumikhin theorem in [3] which states that if there exist a continuous Lyapunov function V(t, x), three K-functions u, v, w and a continuous nondecreasing function q(s) > s for s > 0, such that the following two conditions are satisfied for all t ∈ [t0, ∞):(1)u(x)V(t,x)v(x),forall(t,x)R+×Rn;(2)V˙(t,x(t))w(x(t)),ifV(t+θ,x(t+θ))q(V(t,x(t))),θIτ, then the trivial solution of the Eq. (1) is globally uniformly asymptotically stable.

By the above Razumikhin theorem, in order to guarantee asymptotic stability of the Eq. (1), we need to find a positive-definite function V(t, x) whose time-derivative V˙(t,x(t)) along the solution of the Eq. (1) is negative-definite under a Razumikhin-type condition. It is shown that the asymptotic behavior or stability properties of the solution of the Eq. (1) can be guaranteed by certain negative-definite conditions of the derivative of Lyapunov functions such as V˙(t,x(t))w(x(t)), under some Razumikhin-type conditions such as V(t+θ,x(t+θ))q(V(t,x(t))), for all θ ∈ Iτ. In other words, function w( · ) is used to give qualitative properties of the solutions of functional differential equations; see [10], [17], [18]. Actually, the principle of the Razumikhin technique is to determine qualitative properties using the condition (2) above. The above condition (2) will be stated formally as a Razumikhin-type functional differential inequality in the sequel.

On the other hand, one may feel intuitively that different w(‖x(t)‖) in the condition (2) above implies different decay rate for the solution of the Eq. (1). The larger w(‖x(t)‖) is, the larger the decay rate for the solution of the Eq. (1) should be. That is, an inequality as V˙(t,x(t))w(x(t)) under some Razumikhin-type conditions may imply some quantitative information for the solution x(t). This means that the so-called Razumikhin-type functional differential inequality stated in this paper can provide both quantitative description and qualitative description for the solutions of functional differential equations.

In general, what kinds of quantitative properties can be obtained depends on the form of the Razumikhin-type functional differential inequality. There usually exist three common forms of the negative-definite conditions of V˙(t,x(t)) along the solution x(t) of the Eq. (1) in the corresponding Razumikhin-type functional differential inequality. One form is V˙(t,x(t))w(x(t)); see [3], another one is the form of V˙(t,x(t))w(V(t,x(t))); see [19], [20]. Actually, w(‖x(t)‖) and w(V) are equivalent in essence under some simple conditions, which will be proved in Theorem 2 of this paper. So we only refer to w(V) in the following text. The last form involves two variables and is denoted by a function w¯(t,V), i.e., V˙(t,x(t))w¯(t,V(t,x(t))); see [4], [5], [6], [10], [13], [17], [18].

For the special case with w(V)=αV, for α > 0, the exponential stability of stochastic functional differential equations was easily obtained; see [19], [20]. Later, [12], [21], [26] established the Razumikhin-type theorems to a time-varying case and extended the exponential stability to the general decay stability of stochastic functional differential equations. Ning et al. [13] proposed an improved Razumikhin-type stability theorems for input-to-state stability of nonlinear time-delay systems. Li et al. [24] focused on the finite-time stability of time-varying time-delay systems by the Razumikhin technique and weakened the negative-definite condition for V˙.

The time-varying case with w¯(t,V)=α(t)β(V) (where β(V) ≥ 0 for all V > 0 and α(t) ≥ 0 for all t > 0) and its transformation were extended, and the decay estimate for applications of Razumikhin-type theorems and criteria for quantitative stability for a class of Razumikhin-type retarded functional differential equation were obtained; see [4], [5], [6]. Li and Song [14], Cheng et al. [28], Liu and Yang [29] relaxed the non-negative condition for α(t) to a more general case, and proposed new extensions of Razumikhin-type stability theorems and applied them to impulsive models. Teel [7] established the connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. Later, Ning et al. [13] proposed an improved Razumikhin-type stability theorems for input-to-state stability of nonlinear time-delay systems and also weakened the negative-definite condition of V˙. Zhou and Egorov [11] generalized the results of [13] to a weaker one and obtained Razumikhin stability of time-varying time-delay systems.

In this aspect, we can see that the negative-definite condition of V˙ in the existing literature admits a general case that time t and the Lyapunov function V are separated with each other. And the generalization of the negative-definite condition for V˙ has been received much more attention; see [11], [13], [14], [24], [34]. However, little work has been done for the case that time t and the Lyapunov function V are inseparable in the negative-definite condition of V˙. Recently, [10], [17], [18] considered the function w¯(t,V) with the case that time t and the Lyapunov function V are inseparable and obtained some novel asymptotical stability theorems for (hybrid) stochastic retarded systems by the Razumikhin technique. To the best of the authors’ knowledge, the general result on quantitative properties of the Razumikhin-type functional differential inequalities has been barely systematically studied.

Motivated by the above analysis, we would like to investigate quantitative properties of the Razumikhin-type functional differential inequalities under a more relaxed condition. In this paper, general and uniform bounds for general Razumikhin-type functional differential inequalities are established. We will further show how to bound the solutions of the Razumikhin-type functional differential inequalities under the general case that time t and the Lyapunov function V are inseparable. Based on these bounds, we will present another approach to deduce some basic Razumikhin-type stability results, which are more direct than those reported so far in the related literature.

The structure of the paper is as follows: The next section introduces some preliminaries. A lemma for bounding Razumikhin-type functional differential inequalities is also presented. In Section 3, some quantitative bounds of the basic Razumikhin-type functional differential inequalities are established. In Section 4, a Razumikhin-type stability theorem for deterministic functional differential equations is provided. In Section 5, Razumikhin-type stability theorems for stochastic functional differential equations are studied in detail. Finally, we illustrate our method using two examples in Section 6 and give a conclusion in Section 7.

Section snippets

Notations

Throughout the paper, (Ω,F,{Ft}tt0,P) is a complete probability space with a filtration {Ft}tt0 satisfying the usual conditions, i.e. it is right continuous and Ft0 contains all P-null sets. ‖ · ‖ is the vector norm. τ is a positive constant which stands for the upper bound for the bounded time-delay involved possibly in the inequalities or equations. Let t0R+=[0,+),R=(,0], and T be a positive constant with T > t0 ≥ 0, or infinity. I=[t0τ,T) is the existing interval for the solutions of

Bounds for Razumikhin-type functional differential inequalities

Based on the above bound, we derive the quantitative bounds for Razumikhin-type functional differential inequalities, which are described by some integral inequalities, without any Razumikhin condition. Based on quantitative bounds, we can obtain stable or asymptotic properties of the solutions of Razumikhin-type functional differential inequalities. These bounds are the essential foreshadowing for us to deduce the basic Razumikhin-type stability theorems in the next section.

Theorem 1

Under Assumption 1,

Application to stability of deterministic functional differential equations

In this section, we will demonstrate the application of the above obtained bounds to a classical Razumikhin-type stability theorem for deterministic functional differential equations. This will be compared with the methods employed to establish Razumikhin-type stability theorems in the existing literature; see [3], [19]. The method of using our bounds turns out to be more direct and distinctly different from those used in the existing literature.

Consider the deterministic functional

Application to stability of stochastic functional differential equations

Consider the following Ito^ stochastic functional differential equation{dx(t)=f(t,xt)dt+g(t,xt)dw(t),tt0,xt0(θ)=ϕ0(θ),θIτ, where xRn, f:R+×C(Iτ;Rn)Rn and g:R+×C(Iτ;Rn)Rn×m are assumed to be measurable functionals with f(t,0)=0Rn,g(t,0)=0Rn×m for all t ≥ t0. w(t) is an m-dimensional standard Wiener process defined on the complete probability space (Ω,F,{Ft}tt0,P). The initial condition for the Eq. (7) will given by (t0, ϕ0), where ϕ0={ϕ0(s):sIτ} is an Ft0-measurable C(Iτ;Rn)-valued

Illustrative examples

In this section, we consider two models described by a deterministic functional differential equation and a stochastic functional differential equation to verify the effectiveness of our established theorems, respectively.

Example 1

Consider a two-dimensional deterministic differential equation with a time-varying delay{x˙(t)3mm=A(t)x(t)+B(t)x(tτ(t)),tt0,xt0(θ)3mm=ϕ0(θ),θIτ, where AC(R+;R2×2),BC(R+;R2×2) are continuous matrix satisfyingA(t)=[4+2sint+2t1t+11t+15+2sint+2t],B(t)=[sint+t1212sint+t],

Conclusion

In this paper, we explicitly propose the notion of Razumikhin-type functional differential inequalities and establish the fundamental results on the quantitative bounds for the Razumikhin-type functional differential inequalities. By these quantitative bounds, Razumikhin-type stability results are deduced for both deterministic functional differential equations and Ito^ stochastic functional differential equations. It should be pointed out that the quantitative bounds established in this paper

CRediT authorship contribution statement

Xueyan Zhao: Conceptualization, Methodology, Software, Writing - original draft. Minyue Fu: Writing - review & editing. Feiqi Deng: Writing - original draft, Formal analysis, Validation, Writing - review & editing. Qigui Yang: Supervision.

Declaration of Competing Interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Acknowledgements

The authors would like to thank the National Natural Science Foundation of China under Grants 61873099, U1701264, 61633014, 61733008 and the Natural Science Foundation of Guangdong Province Under Grant 2020A1515010441 and Guangdong Provincial Key Laboratory of Technique and Equipment for Macromolecular Advanced Manufacturing (SCUT) for their financial support.

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