Stability analysis for nonlinear Markov jump neutral stochastic functional differential systems

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Highlights

  • The rigorous constraint of global Lipschitz condition for the delay parts of the drift coeffcients is weakened to our Assumption 5, which can be specially applicable to the situation of the high nonlinearity for delay parts of the drift coeffcients.

  • Except for the asymptotic stability, the matter of exponential decay speed is creatively investigated by designing a new type of functionals, whose criteria depend on the intervals lengths of continuous delays.

  • The degenerate functionals are triumphantly imported into such complicated stochastic systems to survey the stability property.

Abstract

Recently, the asymptotic stability for Markov jump stochastic functional differential systems (SFDSs) was studied, whose stability criteria relied on the intervals lengths of continuous delays. Whereas, so far all the existing references require the rigorous global Lipschitz condition for the delay parts of the drift coefficients and do not consider the challenging factors of exponential decay and neutral issue. Motivated by the aforementioned considerations and the advantages of the degenerate functionals, this paper aims to weaken the global Lipschitz condition for the delay parts of the drift coefficients and investigate the delay-dependent exponential stability and asymptotic boundedness for highly nonlinear Markov jump neutral SFDSs with the method of multiple degenerate functionals. Of course, the delay-independent assertions are as well derived here.

Introduction

Stochastic functional/delay differential systems (SFDSs/SDDSs) have been universally acknowledged as efficient models to simulate practical systems with random factor and time delays factor in the continuous/discrete form (e.g. [1], [2]). In reality, sudden variations are commonly underwent now and then. The Markov chain is a shared approach to deal with this problem (e.g. [3], [4]). Thereby, Markov jump SFDSs/SDDSs are established to model these systems with sudden variations. To put it from another angle, these jump systems can be thought of an array of subsystems switching mutually according to the transition rates given by the Markov chain. For SFDSs or Markov jump SFDSs, considerable focuses are poured into the stability property, and we just mention the references [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], among the others. Thereinto, the criteria on the stability for SFDSs or Markov jump SFDSs are classified into the delay-dependent case and the delay-independent case. Here we specially emphasize on the works [21], [22]: the authors mainly probed the asymptotic stability and H stability for highly nonlinear Markov jump SFDSs, where the stability criteria were dependent on the intervals lengths of continuous delays.

A non-ignorable fact that systems derivates are as well affected by continuous delays on some finite intervals besides of discrete delays, must be revealed out. Consequently, Markov jump neutral SFDSs (NSFDSs) are imported to meet the above requirement. Then, growing scholars begin to turn their emphasis toward the stability matter of Markov jump NSFDSs (e.g. [23], [24], [25], [26], [27]). Whereas, so far there are little literature concerning the delay-dependent stability criteria for Markov jump NSFDSs. Stimulated by the thoughts of the works [21], [22], it is a spontaneous consideration to examine the delay-dependent stability for highly nonlinear Markov jump NSFDSs.

It must be alluded out that, in a sense, it is trivial to straightforward move the theory of works [21], [22] to Markov jump NSFDSs unless some deep-seated factors are further taken into account, such as the following two aspects: 1) The rigorous global Lipschitz condition is indispensable for the delay parts of the drift coefficients in works [21], [22], which will lose efficacy for many situations; 2) The chief assertions of asymptotic stability and H stability in works [21], [22] lose sight of the significant matter of convergent speed. Therefore, to highlight the non-trivial contributions of this paper, the two aspects above will be further developed.

What is more, a class of degenerate functionals were designed by the works [28], [29] to probe the stability of SDDSs with the linear growth condition, and the method was further developed to study the functional systems by the work [30]. More importantly, degenerate functionals weaken the virtual requirement of nonnegative definite lower boundedness of Lyapunov functions/functionals employed by all the literature mentioned above. Then, a challenging problem arises: whether the method of degenerate functionals can be applied for highly nonlinear Markov jump NSFDSs? In this case, unquestionably, the synchronous emergence of high nonlinearity, Markov jump and neutral issue will aggravate the analysis trouble of adopting the degenerate functionals to examine the stability issue.

Summing up the aforesaid and drawing the mind from work [31], this paper will focus on the delay-dependent exponential stability and asymptotic stability for highly nonlinear Markov jump NSFDSs by constructing different degenerate functionals for different subsystems. Concretely speaking, multiple degenerate functionals are assumed to be controlled by two functions and their single integrals, and the corresponding Itô’s operators are assumed to be governed by other functions, their single and double integrals. The principal highlights of this paper relative to the state-of-the-art are concentrated on the following perspectives:

  • The rigorous constraint of global Lipschitz condition for the delay parts of the drift coefficients is weakened to our Assumption 5 can be specially applicable to the situation of the high nonlinearity for delay parts of the drift coefficients;

  • Except for the asymptotic stability, the matter of exponential decay speed is creatively investigated by designing a new type of functionals, whose criteria depend on the intervals lengths of continuous delays;

  • The degenerate functionals are triumphantly imported into such complicated stochastic systems to survey the stability property.

Section snippets

Problem description

Let G={1,2,,N}, δ>0, R=(,+), R+=[0,+), Rn be the space of n-dimensional Euclidean space with the Euclidean norm |·|, Rn×m be the space of n×m-matrixes, C(S1;S2) be the class of all continuous functions η:S1S2 (particularly, C([δ,0];Rn) be with the norm η=supδs0|η(s)|), and μi be the probability measures on [δ,0] (i.e., δ0dμi(ρ)=1), i=0,1,,9. MT and |M|=trace(MTM) represent the transpose and trace norm of matrix M. Denote (Ω,F, {Ft}t0,P) by a complete probability space with the

Main results

To pledge the existence and explore the stability property of the unique global solution ξζ,Λ0(t) of nonlinear Markov jump NSFDS (2.1), we set ξ(t)=ξ(δ) for t[2δ,δ), Λ(t)=Λ(0) for t[2δ,0), f(φ(0),φ,t,k)=f(φ(0),φ,0,k) and g(φ(0),φ,t,k)=g(φ(0),φ,0,k) for (φ,t,k)C([δ,0];Rn)×[2δ,0)×G, construct different degenerate functionals for different subsystems, and propose some more assumptions.

Assumption 3

There are some functions VC2,1(C([δ,0];Rn)×R+×G;R+), FjC(Rn×[δ,);R+),j=0,1,,3, F4C(Rn×[2δ,);R+

An example

Here, we give an example to illustrate the effectiveness of our theory.

Example 1

See a scalar Markov jump NSFDSd[ξ(t)U(ξt)]=f(ξ(t),ξt,t,Λ(t))dt+g(ξ(t),ξt,t,Λ(t))dB(t),where U(ξt)=0.020.0050ξ(t+ρ)dμ8(ρ), B(t) is a scalar Brownian motion, Λ(t) is a Markov chain with the generator Λ=(0.50.566) and G={1,2}, andf(ξ(t),ξt,t,1)=6ξ3(t)+16[0.0050ξ(t+ρ)dμ8(ρ)]250.0050ξ(t+ρ)dμ8(ρ),f(ξ(t),ξt,t,2)=4ξ3(t)+16[0.0050ξ(t+ρ)dμ8(ρ)]2,g(ξ(t),ξt,t,1)=130.0050ξ(t+ρ)dμ8(ρ),g(ξ(t),ξt,t,2)=140.0050ξ(t+ρ)dμ8(ρ).

Conclusions

In a summary, the method of multiple degenerate functionals is successfully applied to investigate the stability and boundedness issues for highly nonlinear Markov jump NSFDSs. Besides of the delay-independent stability assertions, this paper creatively concentrates on the exponential stability whose criteria rely on the intervals lengths of continuous delays, which is the main contribution. Nevertheless, this method is mainly applicable for the finite delays case. Up to now, we do not know how

Acknowledgments

The authors would like to thank National Natural Science Foundation of China (Nos. 11571024, 12071102, 61773152), China Postdoctoral Science Foundation (No. 2017M621588), Natural Science Foundation of Hebei Province of China (No. A2019209005), Science and Technology Research Foundation of Higher Education Institutions of Hebei Province of China (No. QN2017116), Tangshan Science and Technology Bureau Program of Hebei Province of China (No. 19130222g) and Hebei Provincial Postgraduate

References (35)

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