Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 9097135, 9 pages
https://doi.org/10.1155/2019/9097135

Research Article

On the Novel Finite-Time Stability Results for Uncertain Fractional Delay Differential Equations Involving Noninstantaneous Impulses

Qien Li, Danfeng Luo, Zhiguo Luo, and Quanxin Zhu

Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

Correspondence should be addressed to Danfeng Luo; luodf0916@sohu.com

Received 22 July 2019; Revised 10 September 2019; Accepted 16 September 2019; Published 29 September 2019

Academic Editor: Ines Tejado Balsera

Copyright © 2019 Qien Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we mainly study the finite-time stability for a kind of uncertain fractional-order delay differential equations with noninstantaneous impulses. By using the Lyapunov functions along with the generalized Grönwall inequality, we present the new stability results for the considered equations. Finally, two examples are given to demonstrate the effectiveness of our theoretical results.

1. Introduction

Over the past two decades, fractional-order systems (FOSs) have been intensively studied due to its wide applications to various fields, especially in memory storage, dynamics mechanic, and communication engineering. As we know, stability theory is one of the most crucial themes in the qualitative theory of dynamical systems, and we can see the monographs [1, 2] and the papers [3–7] for Ulam–Hyers stability, [8–11] for Lipschitz stability, [12–14] for asymptotic stability, and for other forms of stability. Further, for more details about some other properties of the solutions, we can see [15–18]. In this paper, we mainly study the finite-time stability, which has been investigated by some researchers. For an extensive collection of such results, we refer the readers to the related literatures, such as the papers [19–23]. In detail, in [19], Wu et al. studied the finite-time stability of Caputo delta fractional linear difference equations with the aid of Grönwall inequality. In [20], D. F. Luo and Z. G. Luo considered the uniqueness and finite-time stability of solutions for a kind of fractional-order nonlinear difference equations with time-varying delays and improved the results in [19]. In order to explore the finite-time stability of fractional differential equations, Li and Wang [21] introduced a concept of delayed Mittag-Leffler type matrix function and studied the system in the following form:

The authors finally presented the finite-time stability results by virtue of delayed Mittag-Leffler type matrix.

Phat and Thanh [22] considered the following nonlinear FOSs with time-varying delay and nonlinear disturbance:

Based on a generalized Grönwall inequality, a delay-dependent sufficient condition for robust finite-time stability of such systems was established in terms of the Mittag-Leffler function.

To research the influence of impulses on finite-time stability of fractional differential system, D. F. Luo and Z. G. Luo in [23] investigated the following ψ-Hilfer fractional differential equation with time-varying delays and noninstantaneous impulses:

where , is the ψ-Hilfer fractional derivative of order and type , with respect to function ψ, , is the quantity of state mapping the interval J to , , are bounded operators, is continuous and , , and is a continuous delay function satisfying , , is a delayed function. The authors deduced that the solutions of system (3) had finite-time stability under some new criteria and by applying the generalized Grönwall inequality.

In this paper, we first assume two increasing finite sequences of points and are given, such that , where is given and is a natural number. Inspired by the papers mentioned, we consider the following uncertain fractional-order delay differential equations with noninstantaneous impulses:

where , is the Caputo fractional derivative of order , and is a continuous function, such that , , . Let be a given function to be specified later, and the uncertainty of this system denoted by , which is bounded in . are continuous maps for , are noninstantaneous impulsive functions for all , and is a continuous function.

Remark 1 [23]. If , , then system (4) reduces to an impulsive differential equation. In this case, at any point of instantaneous impulse , the amount of jump of the solution is given by .
The main objective in this paper is to extend the work in [24, 25] to fractional order and develop the results in [21, 22] under noninstantaneous impulsive effects, where impulsive action starts at an arbitrary fixed point and remains active on a finite time interval, which is very different from the classical instantaneous impulsive case that changes are relatively short compared to the overall duration of the whole process. Our main motivation for doing this paper is to research the effect of noninstantaneous impulses on finite-time stability, and to do this, we introduce two class- functions, γ and β, that satisfy some certain conditions, which can guarantee the finite-time stability. The difficulty of the whole process is how to construct γ and β that meet the particular properties we wanted.
This article is organized as follows. In Section 2, we introduce some known definitions and lemmas, which are useful to our following works. Section 3 is devoted to researching the finite-time stability for fractional system (4). To explain the results clearly, we finally provide two examples in Section 4.

2. Preliminaries

In this section, we plan to introduce some basic definitions and lemmas which are used throughout this paper.

Definition 1 [23]. One-parameter Mittag-Leffler function: the Mittag-Leffler function is given by the following series:

where , , and is a gamma function given by

. In particular, if , we have .

Lemma 1 [22]. Generalized Grönwall inequality: suppose that , , is a nonnegative function locally integrable on , is a nonnegative, nondecreasing continuous function defined on , is a nonnegative locally integrable function on satisfying the inequality, the we get

then

Moreover, if is a nondecreasing function on , then for , we have , where is the Mittag-Leffler function defined by Definition 1.

Definition 2 [26]. The Riemann–Liouville fractional integral of order for a continuous function is defined as

where denotes the Riemann–Liouville fractional integral of order α and is Euler’s Gamma function satisfying .

Definition 3 [26]. The fractional-order derivative in the Caputo sense for a continuous function is defined as

where n is the smallest integer greater than or equal to α.

Lemma 2 [5]. If , then the following is satisfied:

where , . In particular, and , then .

Definition 4. A continuous function is said to belong to class- if it is strictly increasing and .
Now, we introduce a space:

with the norm , . Obviously, is a Banach space.

Definition 5 [23]. The fractional system (4) is finite-time stable w.r.t. , , if and only if

Lemma 3. Let , as for any , x satisfies the fractional system (4) if and only if x satisfies the following Volterra integral equations:

Proof. The proof process can be referred to [23].

3. Main Results

Let denote the set of all nonnegative bounded functions on interval J. To establish the main results, we introduce the following assumptions:

  For any , assume that the uncertainty term satisfies the following condition:

where is nonnegative continuous function.  Assume that the nonlinear function f satisfies: there exists a positive function such that

where , , and .  For , there exists a positive sequence , such that

and holds.

Throughout this paper, we always assume the following:

We denote

Based on the known results [21–23], we can clearly conclude that this fractional system (4) has at least one solution under some conditions. Now, we plan to investigate the specified stability.

Theorem 1. Suppose the validity of conditions – and there exist two positive real numbers δ and σ such that , and , then fractional system (4) has finite-time stability on J provided that the condition

holds, where , and .

Proof. The proof process can be referred to Theorem 4.1 in [23].
Now, we need to take Lyapunov function approach to deal with the finite-time stability. In order to accomplish this work, we first consider the following uncertain fractional-order delay differential equations without any impulses:

Lemma 4. Fractional system (21) has finite-time stability provided that there exists a continuously differentiable nonnegative function , a positive nondecreasing continuous function , and two nonnegative continuous functions , defined on J, and two class- functions, γ and β, such that for any ,

(1)(2)(3)(4)δ and σ defined as in Definition 5 meet the following inequality: (5) and are both locally integrable on

Proof. The analysis of Definition 2 shows that operator is nondecreasing and linear. We apply the fractional integral operator to the both sides of condition and by Lemma 2, then we have

where . Obviously, we can further get by condition the following:

Then

Let , , and by condition (5), we know that and satisfy the conditions in Lemma 1. Hence, one can obtain

which implies that

According to the condition , we can get

which implies that system (21) has finite-time stability. The proof of this Lemma is completed.

Theorem 2. Suppose the validity of – and are continuous linear maps for , such that , , and there exist two positive real numbers δ and σ such that and , then fractional system (4) has finite-time stability if conditions (1)–(3) and (5) in Lemma 4 and

hold, where , , , and is defined in the following expression (49).

Proof. For , we have , and it is obvious that system (4) has finite-time stability on .
As , the proof process is similar to the proof in Lemma 4.
As , we also apply the fractional integral operator on the both sides of condition (2), then we have

where is called the initial value on , and

According to expression (14), for any , one can obtain

where is defined in the same way as in Theorem 1. We refer to the idea in [23] and set , and , then , , and . From (31), it follows

Noting that for all , we can get

Hence, we obtain the following result:

By the generalized Grönwall inequality in Lemma 1, we have

which implies that, for any ,

From (30) and (36), we can conclude that

By using the same idea in Lemma 4, we can easily get

and we can see the fractional system (4) has finite-time stability on by the Definition 5.
As , we also apply the fractional integral operator on the side of condition (2), then we have

where is called the initial value, and

Therefore, we easily obtain by Lemma 4 the following:

and we can see the fractional system (4) has finite-time stability on by the Definition 5.
As , we have the initial value:

By the Volterra integral equation (14), for all , we have

Using the same approach in (36), one can get

By (42) and (44), we can obtain

Therefore, we easily obtain by Lemma 4 the following:

and we can see that the fractional system (4) has finite-time stability on by the Definition 5.
And so on and so forth, we get the following results on the other intervals. As , , we have

which implies that the fractional system (1.4) has finite-time stability on , by the Definition 5.
As , , we have

Therefore, we can see that the fractional system (4) has finite-time stability on by the Definition 5. The proof of this theorem is completed.

Remark 2. System (4) will be an instantaneous impulsive differential equation under , , then the finite-time stability is affected by the number of impulses, which has been a very important and meaningful research topic, and investigated by some experts [27–29]. The results in this literature are new and can improve some referred conclusions.

4. Examples

In this section, we will present the following two examples to illustrate our main results.

Example 1. Assume that , and let , , , , and . Let be a natural constant, , , then , , , , and , where . Consider , and by Mathematica software, we have

To make sure that inequality (20) is true, and , we only need to make not greater than 2.05, which is easy to be done. Let , , and . All conditions in Theorem 1 are satisfied, and system (4) has finite-time stability on .

Example 2. All the conditions are the same as in above example unless otherwise stated. Let the Lyapunov function be , then one can obtain

Assume that , . It is obvious that

Let , , and . By calculating, we can choose . By Mathematica software, we have

and we choose . All the conditions in Theorem 2 can be verified. Hence, the fractional system (4) has finite-time stability.

Remark 3. Since there are few papers researching the finite-time stability of solutions for the uncertain fractional-order nonlinear differential equation involving time-varying delays and noninstantaneous impulses, one can see that all the results in [21, 22, 24, 25, 27–29] cannot directly be applicable to the examples mentioned above to obtain the results. This implies that the results in this paper are essentially new.

5. Conclusions

In this paper, we are concerned with a class of Caputo fractional-order differential equations. The addressed equation has noninstantaneous impulsive effects, which are quite different from the related references discussed in the literature [21, 22]. The nonlinear fractional-order differential system studied in the present paper is more generalized and more practical. By the definition of finite-time stability, the easily verifiable sufficient conditions have been provided to determine the stability conclusions. Finally, two typical numerical examples of the stability have been presented at the end of this paper to illustrate the effectiveness and feasibility of the proposed criterion. Consequently, this paper shows theoretically and numerically that some related references known in the literature can be enriched and complemented.

An interesting extension of our study would be to discuss finite-time stability for the fractional differential equations with discontinuous right-hand side. This topic will be the subject of a forthcoming paper.

Data Availability

The data in this study were mainly collected via discussion during our class. Readers wishing to access these data can do so by contacting the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research was supported by the Hunan Provincial Innovation Foundation for Postgraduate (grant no. CX2018B288) and the National Natural Science Foundation of China (grant no. 11471109).

References

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies 204, vol. 207, Elsevier, Amsterdam, Netherlands, 2006.

K. S. Miller and B. Rose, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience Publication, Hoboken, NJ, USA, 1993.

J. V. C. Sousa and E. C. Oliveira, “On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator,” Journal of Fixed Point Theory and Applications, vol. 20, no. 3, p. 96, 2018.

E. C. de Oliveira and J. V. C. da Sousa, “Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations,” Results in Mathematics, vol. 73, no. 3, p. 111, 2018.

H. Khan, T. Abdeljawad, M. Aslam, R. A. Khan, and A. Khan, “Existence of positive solution and Hyers-Ulam stability for a nonlinear singular-delay-fractional differential equation,” Advances in Difference Equations, vol. 2019, no. 1, p. 104, 2019.

K. Shah, A. Ali, and S. Bushnaq, “Hyers–Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions,” Mathematical Methods in the Applied Sciences, vol. 41, no. 17, pp. 8329–8343, 2018.

D. F. Luo, K. Shah, and Z. G. Luo, “On the novel Ulam–Hyers stability for a class of nonlinear ψ-Hilfer fractional differential equation with time-varying delays,” Mediterranean Journal of Mathematics, vol. 16, no. 5, p. 112, 2019.

F. M. Dannan and S. Elaydi, “Lipschitz stability of nonlinear systems of differential equations,” Journal of Mathematical Analysis and Applications, vol. 113, no. 2, pp. 562–577, 1986.

S. Hristova and R. Terzieva, “Lipschitz stability of differential equations with non-instantaneous impulses,” Advances in Difference Equations, vol. 2016, no. 1, p. 322, 2016.

S. Hristova and K. Ivanova, “Lipschitz stability of delay differential equations with non-instantaneous impulses,” Dynamic Systems & Applications, vol. 28, no. 1, pp. 167–181, 2019.

R. Agarwal, S. Hristova, and D. O’Regan, “Lipschitz stability for non-instantaneous impulsive Caputo fractional differential equations with state dependent delays,” Axioms, vol. 8, no. 1, p. 4, 2019.

M. Fečkan and J. R. Wang, “Periodic impulsive fractional differential equations,” Advances in Nonlinear Analysis, vol. 8, no. 1, pp. 482–496, 2017.

N. Cong, T. Doan, S. Siegmund, and H. Tuan, “Linearized asymptotic stability for fractional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 39, no. 39, pp. 1–13, 2016.

D. Baleanu, G. C. Wu, and S. D. Zeng, “Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations,” Chaos, Solitons & Fractals, vol. 102, pp. 99–105, 2017.

V. H. Ngo, “Fuzzy fractional functional integral and differential equations,” Fuzzy Sets and Systems, vol. 280, pp. 58–90, 2015.

S. Salahshour, A. Ahmadian, M. Salimi, M. Ferrara, and D. Baleanu, “Asymptotic solutions of fractional interval differential equations with nonsingular kernel derivative,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29, no. 8, Article ID 083110, 2019.

A. Ahmadian, C. S. Chan, S. Salahshour, and V. Vaitheeswaran, “FTFBE: a numerical approximation for fuzzy time-fractional Bloch equation,” in Proceedings of the 2014 IEEE international conference on fuzzy systems (FUZZ-IEEE), pp. 418–423, Beijing, China, July 2014.

A. Ahmadian, S. Salahshour, M. Ali-Akbari, F. Ismail, and D. Baleanu, “A novel approach to approximate fractional derivative with uncertain conditions,” Chaos, Solitons & Fractals, vol. 104, pp. 68–76, 2017.

G. C. Wu, D. Baleanu, and S. D. Zeng, “Finite-time stability of discrete fractional delay systems: gronwall inequality and stability criterion,” Communications in Nonlinear Science and Numerical Simulation, vol. 57, pp. 299–308, 2018.

D. F. Luo and Z. G. Luo, “Uniqueness and novel finite-time stability of solutions for a class of nonlinear fractional delay difference systems,” Discrete Dynamics in Nature and Society, vol. 2018, Article ID 8476285, 7 pages, 2018.

M. Li and J. Wang, “Finite time stability of fractional delay differential equations,” Applied Mathematics Letters, vol. 64, pp. 170–176, 2017.

V. N. Phat and N. T. Thanh, “New criteria for finite-time stability of nonlinear fractional-order delay systems: a Gronwall inequality approach,” Applied Mathematics Letters, vol. 83, pp. 169–175, 2018.

D. F. Luo and Z. G. Luo, “Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses,” Advances in Difference Equations, vol. 2019, no. 1, p. 155, 2019.

M. Radosław and R. Andrzej, “Global finite-time stability of differential equation with discontinuous right-hand side,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 35, pp. 1–17, 2018.

M. Anna and N. Andrzej, “Finite-time stability and finite-time synchronization of neural network-dual approach,” Journal of the Franklin Institute, vol. 354, no. 18, pp. 8513–8528, 2017.

P. Eloe and J. Neugebauer, “Concavity in fractional calculus,” Filomat, vol. 32, no. 9, pp. 3123–3128, 2018.

F. Amato, G. De Tommasi, and A. Pironti, “Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems,” Automatica, vol. 49, no. 8, pp. 2546–2550, 2013.

F. Amato, R. Ambrosino, M. Ariola, and G. De Tommasi, “Robust finite-time stability of impulsive dynamical linear systems subject to norm-bounded uncertainties,” International Journal of Robust and Nonlinear Control, vol. 21, no. 10, pp. 1080–1092, 2011.

R. Ambrosino, F. Calabrese, C. Cosentino, and G. De Tommasi, “Sufficient conditions for finite-time stability of impulsive dynamical systems,” IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 861–865, 2009.