Open access peer-reviewed chapter

Existence, Uniqueness and Stability of Fractional Order Stochastic Delay System

Written By

Sathiyaraj Thambiayya, P. Balasubramaniam, K. Ratnavelu and JinRong Wang

Reviewed: February 14th, 2022 Published: April 2nd, 2022

DOI: 10.5772/intechopen.103702

Chapter metrics overview

33 Chapter Downloads

View Full Metrics

Abstract

This chapter deals with the problem of fractional higher-order stochastic delay systems. A solution representation is given by using sin and cos matrix functions for different delay intervals. Further, existence and uniqueness results are proved through fixed point theorem. Moreover, finite-time stability criteria are obtained using fractional Gronwall-Bellman inequality lemma. Finally, numerical simulation is carried out to check the proposed theoretical results.

Keywords

  • existence and uniqueness of solution
  • fixed point theorem
  • fractional differential equations (FDEs)
  • stochastic differential system

1. Introduction

Fractional derivatives (FD) initiative concept is quite old and its history spans three centuries. The variety of papers dedicated to FD is multiplied swiftly in the mid-twentieth century and later decades. One of the motives for the full-size interest within the discipline of FD is that it’s far feasible to describe the variety of physical [1], synthetic [2], and organic [3] occurrence with fractional differential equations (FDEs). As a new branch of applied mathematics, the field of FD can be seen in many applications. Nevertheless, more and more compelling implementations have been found in various engineering and science fields over the past few decades (see [4]). It is noted that the existing theory of FDEs is committed to a larger part of the research works (see [5, 6, 7, 8]). While modeling functional structures, ambient noise and time delays need to be taken into account, which might be very beneficial in building extra sensible fashions of sciences, and so on [9]. It is referred to as the pattern direction houses of the stochastic fractional partial differential gadget powered by way of area time white noise [10].

The problems in a stochastic environment replicate the modeling of single-sever m-mode random queues in computer networks [11], the spatial distribution of mobile users in the telecommunications network coverage area [12], and other anomalies that occurred naturally in many disciplines [13]. Authors in [14] investigated the existence, uniqueness, and large deviation principle solutions to stochastic evolution equations of jump type. Among the many meaningful properties of stochastic stability results describe the maximum vital feature of fractional order stochastic systems and have been investigated in Refs. [15, 16, 17, 18]. The notion of finite-time stability for fractional stochastic delay systems occurs a matter of course in stochastic control systems. Without any doubt that this type of fractional stochastic stability is most important in both theory and applications.

However, only few introductions and discussions exist on the definition of finite-time stability in stochastic finite space using fixed point theorem approach. Burton [19] started to analyze the stability characters of dynamical systems broadly using fixed point theorems. Subsequently, few authors applied fixed point approach to establish sufficient conditions for stability of the differential systems (see [20, 21, 22, 23, 24]). Based on the above discussions, this chapter provides finite-time stability of the Caputo sense FDEs via fixed point theorems.

The primary contribution of this chapter is defined as follows:

  1. A fractional higher-order stochastic delay system (FSDS) is considered in finite-dimensional stochastic settings.

  2. Weaker hypothesis on nonlinear terms and appropriate fixed-point analysis are utilized to obtain the existence and uniqueness of solution.

  3. A new set of generalized sufficient conditions for finite-time stability of a certain FSDS is established by using Generalized Gronwall-Bellman inequality.

Novelties and challenges of this chapter are described through the subsequent statements:

  1. Finite-time stability analysis for FSDS is new in literature of finite-dimensional fractional stochastic settings.

  2. It is a challenge to tackle the proposed system with a norm estimation on nonlinear stochastic terms as described in this chapter.

  3. It is more complex to verify the weaker assumptions of the system and the derived result is new, has not been analyzed with the existing literature.

  4. Obtained result is proved in stochastic nature with square norm settings.

Organization of this chapter is as follows: system description and preliminaries are provided in Section 2. Existence and uniqueness of solution are provided in Section 3. Finite-time stability result is proved in Section 4 and Section 5 consists of a numerical example.

Notations:CDκ+qrepresent respectively the Caputo derivative with q01;Rnand Rn×nrepresent the ndimensional Euclidean space and n×nreal matrix; Edenotes the mathematical expectation with some probability measure; Ω=LF020bRn;for any yRn,we define the norm

yη=supηκbe2yη2;

define a column-wise matrix sum

M=maxk=1jmk1k=1jmk2k=1jmkjn.

Further, let as define the matrix norm

maxηκ0e2ψη2=ψ2,maxηκ0e2ψη2=ψ2.
Advertisement

2. System description and preliminaries

Consider the following system:

CDκ+qCDκ+qyη+M2yηκ=Fηyη+0ηΔζyζdwζ,η0b,κ>0,yη=ψη,yη=ψη,ηκ0,κ>0,E1

where yηRnis a state vector. Here, MRn×nis taken as a nonsingular matrix. Fis mapping from 0b×Rnto Rnand Δis a mapping from 0b×Rnto Rn×dare nonlinear continuous function and ψC1κ0Rnis an initial value function. wdenotes ddimensional Wiener process.

Definition 2.1.([5]) The Caputo derivative forf:κR,is

CDκ+qfη=1Γ1qκηηζqf'ζ,q01,η>κ,f'η=df.

Definition 2.2.(see[5]) Mittag-Leffler function is

Eq,pu=k=0ukΓkq+pforq,p>0.

In particular, forp=1,

Eq,1θuq=Eqθuq=k=0θkukqΓqk+1,θ,uC.

Definition 2.3.(see[25]) The2kqdegree of polynomial for delayed fractionalcosmatrix is given atη=,k=0,1,

cosκ,qMηq=Θ,<η<κ,I,κη<0,IM2η2qΓ2q+1++1kM2kηk1κ2Γ2kq+1,k1κη<,

where Θand Irepresent the zero and identity matrices.

Definition 2.4.([25]) The2k+1qdegree of polynomial for a delayed fractionalsinmatrix is given atη=,k=0,1,

sinκ,qMηq=Θ,<η<κ,Mη+κqΓq+1,κη<0,Mη+κqΓq+1++1kM2k+1ηk1κ2k+1qΓ2k+1q+1,k1κη<.

We have the following square norm estimations:

  1. cosκ,qMηq2=k=0Mη2qkΓ2kq+12E2qM2η2q2,ηk1κ,k=0,1,2,n

  2. sinκ,qMηq2=k=0Mη+κq2kΓkq+12+k=0M2η+κ2q2kΓ2kq+12+2k1=0k2=0Mη+κqk1Γk1q+1M2η+κ2qk2Γ2k2q+1EqMη+κq2+E2qM2η+κ2q2+2EqMη+κqE2qM2η+κ2q,ηk1κ,k=0,1,2,n.

Definition 2.5.System(1)satisfyingyηψηandyηψηforκη0is finite-time stable in mean square with respect to00bδεκ,if and only ifδ1<δδ>0impliesEyη2<εε>0,η0bwhereδ1=maxψ2ψ2denotes the initial time of observation of the system.

Lemma 2.1.[26](Generalized Gronwall-Bellman inequality) Letvη,bηbe nonnegative and locally integrable on0η<band lethηbe a nonnegative, nondecreasing continuous function defined on0η<b,hηM,and letMbe a real constant,q>0with

vηbη+hη0ηηζq1vζ

and then

vηbη+0ηk=1hηΓqkΓkqηζkq1vζ.

Moreover, ifbηis a nondecreasing function on0b.Then

vηbηEq,1hηΓqηq,η0b,

where Eq,1is the one parameter Mittag-Leffler function.

Assumption 1:Let x,yRn,then we take

supηκbe2xηyη2=Exηyη2.

Lemma 2.2.For a nonsingular matrixM,the solution of the inhomogeneous system is

CDκ+qCDκ+qyη+M2yηκ=fη,η0b,κ>0,yη=ψη,yη=ψη,ηκ0,E2

for zero initial value has the below form:

yη=0ηcosκ,qMηκζqfζ,η0b.

Proof.Consider

yη=0ηcosκ,qMηκζqCζ

where Cζ(unknown) ζ0η.By applying CDκ+qCDκ+qon both sides of the above equation one can obtain

CDκ+qCDκ+qyη=cosκ,qMηqCηM20ηcosκ,qMη2κζqCζ=CηM20ηcosκ,qMη2κζqCζ+M20ηκcosκ,qMη2κζqCζ.

Substitute the above expression into (2), one can get

CηM20ηcosκ,qMη2κζqCζ+M20ηκcosκ,qMη2κζqCζ=fη,

since ηκηcosκ,qMη2κζqCζ=0.Hence the proof. □

Using [25] and Lemma 2.2, the solution of (1) is

yη=cosκ,qMηqψκ+M1sinκ,qMηκqψ0+κ0cosκ,qMηκζqψζ+0ηcosκ,qMηκζqFζyζ+0ηcosκ,qMηκζq0ζΔ(λyλ)dwλ.

Define the nonlinear operator P:RnRnby

Pyη=cosκ,qMηqψκ+M1sinκ,qMηκqψ0+κ0cosκ,qM(ηκζ)qψζ+0ηcosκ,qMηκζqFζ,yζ+0ηcosκ,qMηκζq0ζΔλ,yλdwλ,η0b.
Advertisement

3. Existence and uniqueness results

Theorem 3.1.Assume that Assumption 1 hold. Then the system(1)has a unique solution and following inequality is satisfied

K2E2qM2b+κ2q2eNb1Nb2+4be2Nb12Nb<1.

Proof.Let x,yRn.From Assumption 1 for each η0b,we have

PxηPyη220ηcosκ,qMηκζqFζ,xζFζ,yζ2+20ηcosκ,qMηκζq0ζΔλ,xλΔλ,yλdwλ2.

Multiply by e2on both sides, we get

e2PxηPyη220ηcosκ,qMηκζqe[F(ζxζ)F(ζyζ)]2+20ηcosκ,qMηκζqe0ζ[Δ(λxλ)Δ(λyλ)]dwλ22i+ii.E3

First, we estimate (i):

0ηcosκ,qMηκζqe[F(ζxζ)F(ζyζ)]20ηcosκ,qMηκζq2eNηζ×0ηeNηζe2F(ζxζ)F(ζyζ)2E2qM2b+κ2q20ηeNηζ×0ηeNηζe2F(ηxη)F(ηyη)2E2qM2b+κ2q20ηeNηζ2Exηyη2E2qM2b+κ2q2eNb1Nb2Exηyη2.

By Burkholder-Davis-Gundy inequality and Assumption 1, the estimate for (ii) is given by

0ηcosκ,qMηκζqe0ζ[Δ(λxλ)Δ(λyλ)]dwλ24b0ηcosκ,qMbκζq2e2Nbζe2Δ(ζxζ)Δ(ζyζ)24bE2qM2b+κ2q20be2Nbζe2Δ(ηxη)Δ(ηyη)24bE2qM2b+κ2q2e2Nb12NbExηyη2.

From the above two estimates of (i) and (ii), Eq. (3) becomes

EPxηPyη22E2qM2b+κ2q2eNb1Nb2Exηyη2+8bE2qM2b+κ2q2e2Nb12NbExηyη22E2qM2b+κ2q2eNb1Nb2+4be2Nb12NbExηyη2.

This implies that

EPxηPyη2KExηyη2.

Hence, from statement of the Theorem 3.1, the nonlinear operator (P) is a contraction. Hence the nonlinear operator (P) has a unique solution yRn,which is nothing but solution of Eq. (1). Hence the proof. □

Advertisement

4. Finite-time stability

Theorem 4.1.If Assumption 1 hold and provided that

5K1+M12K3+K22K3K2+κ2K2Eq,15b2q2qK2Γqηqεδ.

Then the system(1)is finite-time stable in mean square.

Proof.By multiplying e2on both sides of the solution of system (1), we derive

e2yη25cosκ,qMηqeψκ2+5M1sinκ,qMηκqeψ02+5κ0cosκ,qMηκζqeψζ2+5|0ηcosκ,qM(ηκζ)qeFζ,yζ|2+50ηcosκ,qMηκζqeNηζe0ζΔλ,yλdwλ25cosκ,qMηq2e2ψκ2+5M12sinκ,qMηκq2e2ψ02+5κ2cosκ,qMη+κq2e2ψη2+50ηηζq1×0ηηζ1qcosκ,qMηκζq2e2|Fζ,yζFζ,0|2+200ηηζq1×0ηηζ1qcosκ,qMηκζq2e2|Δ(ζ,yζΔζ,0|2yη25E2qM2b2q2ψ2+5M12(EqMb+κq2+E2qM2b+κ2q22EqMb+κqE2qM2b+κ2q)ψ2+5κ2E2qM2b+κ2q2ψ2+5b2q2qE2qM2b+κ2q20ηηζq1yζ2+20b2q2qE2qM2b+κ2q20ηηζq1yζ25{K1δ1+M12K3+K22K3K2δ1+κ2K2δ1+b2q2qK20ηηζq1yζ2+4b2q2qK20ηηζq1yζ2}5{K1+M12K3+K22K3K2+κ2K2δ+b2q2qK20ηηζq1yζ2+4b2q2qK20ηηζq1yζ2}.

According to Lemma 2.1, let as take

bη=5K1+M12K3+K22K3K2+κ2K2δ

and

hη=5b2q2qK2.

Moreover, bηis a nondecreasing function on 0b,then

vηbηEq,1hηΓqηqyη25K1+M12K3+K22K3K2+κ2K2δEq,15b2q2qK2Γqηq

Then from the statement of Theorem 4.1, we get

yη2ε.

Hence the system (1) is finite-time stable in mean square. Hence the proof. □

Remark 4.1.By using fixed-point rule, existence and uniqueness of solution, and controllability results have been investigated in[27]. Some well-known results on relative controllability of semilinear delay differential system with linear parts defined by permutable matrices are studied in[28]. In this chapter, we proved some new results of finite-time stability criteria in finite-dimensional space by employing Generalized Gronwall-Bellman inequality and suitable assumption on nonlinear terms.

Advertisement

5. An example

Consider Eq. (1) in the below matrix form:

CD0.75+0.5CD0.75+0.5y1η+0.01y1η0.75=3ηy12η1η+0η(ζy1ζΔ1dB1ζ,y1η=2η,y1η=2,η0.75,0;CD0.75+0.5CD0.75+0.5y2η+0.01y2η0.75=3ηy22η1η+0η(ζy2ζΔ2dBsζ,y2η=4η,y2η=4,η0.75,0,E4

where q=0.5,κ=0.75,Δ1=0.3,Δ2=0.5

A=0.1000.1,Fηyη=3ηy12η1ηe23ηy22η1ηe2.Δηyη=ηy1ηe2σ1dB1ηy2ηe2σ2dB2,ψη=2η4η,ψη=24.

Further, we have the following fractional delayed cosmatrices:

cos0.75,0.65Mη0.65=Θ,<η<0.75,I,0.75η<0,IM2η1.3Γ2.3,0η<0.75,IM2η1.3Γ2.3+M4η0.752.6Γ3.6,0.75η<1.5.E5

From Eq.(5) and using basic calculation, one can get E2qM2b+κ2q2=0.9712,eNb1Nb2=0.2683and 4be2Nb12Nb=1.9004.Using the above-obtained values, one can easily verify that

2E2qM2b+κ2q2eNb1Nb2+4be2Nb12Nb<1.

Hence we verified Theorem 3.1. Further, it is easy to verify that for any xη,yηR2.

e2F(ηxη)F(ηyη)23ηExηyη2
e2Δ(ηxη)Δ(ηyη)20.5ηExηyη2

Hence, Fand Δsatisfies Assumption 1. In Figures 1 and 2, we showed the stable response of the system (4) with fractional order q=0.5and q=0.7, respectively. From the above verification, one can conclude that the system (4) is finite-time stable in mean square.

Figure 1.

The system4is stable atq=0.5.

Figure 2.

The system4is stable atq=0.7.

Advertisement

6. Conclusion

In this chapter, we have derived some meaningful and general results for finite-time stability of nonlinear fractional stochastic delay systems. Existence, uniqueness of solution and stability analysis of FSDS have been proved in finite-dimensional stochastic fractional higher-order differential system. Finally, a numerical simulation test is carried out to validate the obtained theoretical results. Derived result generalizes many existing results with integer and fractional-order systems.

Advertisement

AMS subject classifications (2010)

34A08; 43A15; 37C25; 37A50

References

  1. 1. Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Scientific; 2000
  2. 2. Oldham KB. Fractional differential equations in electrochemistry. Advances in Engineering Software. 2010;41:1171-1183
  3. 3. Magin RL. Fractional calculus models of complex dynamics in biological tissues. Computers and Mathematics with Applications. 2010;59:1586-1593
  4. 4. Ortigueira MD. Fractional Calculus for Scientists and Engineers. New York: Springer Science & Business; 2011
  5. 5. Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: North-Holland Mathematics Studies, Elsevier; 2006
  6. 6. Nieto JJ. Solvability of an implicit fractional integral equation via a measure of noncompactness argument. Acta Mathematics Scientia. 2017;37:195-204
  7. 7. Singh J, Kumar D, Nieto JJ. Analysis of an el nino-southern oscillation model with a new fractional derivative. Chaos, Solitons and Fractals. 2017;99:109-115
  8. 8. Tian Y, Nieto JJ. The applications of critical-point theory discontinuous fractional-order differential equations. Proceedings of the Edinburgh Mathematical Society. 2017;60:1021-1051
  9. 9. Mao X. Stochastic Differential Equations and Applications. Chichester: Horwood Publishing, Cambridge; 1997
  10. 10. Wu D. On the solution process for a stochastic fractional partial differential equation driven by space-time white noise. Statistics and Probability Letters. 2011;81:1161-1172
  11. 11. Seo D, Lee H. Stationary waiting times in m-node tandem queues with production blocking. IEEE Transactions on Automatic Control. 2011;56:958-961
  12. 12. Taheri M, Navaie K, Bastani M. On the outage probability of SIR-based power-controlled DS-CDMA networks with spatial Poisson traffic. IEEE Transactions on Vehicular Technology. 2010;59:499-506
  13. 13. Applebaum D. Levy Processes and Stochastic Calculus. Cambridge: Cambridge University Press; 2009
  14. 14. Rockner M, Zhang T. Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principle. Potential Analysis. 2007;26:255-279
  15. 15. Ahmed E, El-Sayed AMA, El-Saka HAA. Equilibrium points, stability and numerical solutions of fractional-order predatorprey and rabies models. Journal of Mathematics Analysis and Applications. 2007;325:542-553
  16. 16. Gao X, Yu J. Chaos in the fractional order periodically forced complex duffing oscillators. Chaos, Solitons and Fractals. 2005;24:1097-1104
  17. 17. Odibat ZM. Analytic study on linear systems of fractional differential equations. Computers and Mathematics with Applications. 2010;59:1171-1183
  18. 18. Wang J, Zhou Y, Fečkan M. Nonlinear impulsive problems for fractional differential equations and Ulam stability. Computers and Mathematics with Applications. 2012;64:3389-3405
  19. 19. Burton TA, Zhang B. Fractional equations and generalizations of Schaefers and Krasnoselskii’s fixed point theorems. Nonllinear Analysis: Theory, Methods and Applications. 2012;75:6485-6495
  20. 20. Balasubramaniam P, Sathiyaraj T, Priya K. Exponential stability of nonlinear fractional stochastic system with Poisson jumps. Stochastics. 2021;93:945-957
  21. 21. Fečkan M, Sathiyaraj T, Wang JR. Synchronization of Butterfly fractional order chaotic system. Mathematics. 2020;8:446
  22. 22. Ren Y, Jia X, Sakthivel R. The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion. Applicable Analysis. 2017;96:988-1003
  23. 23. Shen G, Sakthivel R, Ren Y, Mengyu L. Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process. Collectanea Mathematica. 2020;71:63-82
  24. 24. Sathiyaraj T, Wang JR, Balasubramaniam P. Ulam’s stability of Hilfer fractional stochastic differential systems. The European Physical Journal Plus. 2019;134:605
  25. 25. Liang C, Wang J, O’Regan D. Representation of a solution for a fractional linear system with pure delay. Applied Mathematics Letters. 2018;77:72-78
  26. 26. Ye H, Gao J, Ding Y. A generalized Gronwall inequality and its application to a fractional differential equation. Journal of Mathematical Analysis and Applications. 2007;328:1075-1081
  27. 27. Sathiyaraj T, Balasubramaniam P. Fractional order stochastic dynamical systems with distributed delayed control and Poisson jumps. The European Physical Journal Special Topics. 2016;225:83-96
  28. 28. Wang J, Luo Z, Fečkan M. Relative controllability of semilinear delay differential systems with linear parts defined by permutable matrices. European Journal of Control. 2017;30:39-46

Written By

Sathiyaraj Thambiayya, P. Balasubramaniam, K. Ratnavelu and JinRong Wang

Reviewed: February 14th, 2022 Published: April 2nd, 2022