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Existence, Uniqueness and Stability of Fractional Order Stochastic Delay System

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

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

DOI: 10.5772/intechopen.103702

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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.


  • 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


define a column-wise matrix sum


Further, let as define the matrix norm


2. System description and preliminaries

Consider the following system:


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


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


In particular, forp=1,


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


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,


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


and then


Moreover, ifbηis a nondecreasing function on0b.Then


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

Assumption 1:Let x,yRn,then we take


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


for zero initial value has the below form:




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


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


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

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


Define the nonlinear operator P:RnRnby


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


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


Multiply by e2on both sides, we get


First, we estimate (i):


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


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


This implies that


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. □


4. Finite-time stability

Theorem 4.1.If Assumption 1 hold and provided that


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


According to Lemma 2.1, let as take




Moreover, bηis a nondecreasing function on 0b,then


Then from the statement of Theorem 4.1, we get


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.


5. An example

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


where q=0.5,κ=0.75,Δ1=0.3,Δ2=0.5


Further, we have the following fractional delayed cosmatrices:


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


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


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.


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.


AMS subject classifications (2010)

34A08; 43A15; 37C25; 37A50


  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