On the Novel Finite-Time Stability Results for Uncertain Fractional Delay Differential Equations Involving Noninstantaneous Impulses
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.
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 , Wu et al. studied the finite-time stability of Caputo delta fractional linear difference equations with the aid of Grönwall inequality. In , 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 . In order to explore the finite-time stability of fractional differential equations, Li and Wang  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  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  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 . 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.
In this section, we plan to introduce some basic definitions and lemmas which are used throughout this paper.
Definition 1 . 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 . 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 getthenMoreover, if is a nondecreasing function on , then for , we have , where is the Mittag-Leffler function defined by Definition 1.
Definition 2 . The Riemann–Liouville fractional integral of order for a continuous function is defined aswhere denotes the Riemann–Liouville fractional integral of order α and is Euler’s Gamma function satisfying .
Definition 3 . The fractional-order derivative in the Caputo sense for a continuous function is defined aswhere n is the smallest integer greater than or equal to α.
Lemma 2 . 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.
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 .
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 thatwhere , , and . For , there exists a positive sequence , such thatand holds.
Throughout this paper, we always assume the following:
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 conditionholds, where , and .
Proof. The proof process can be referred to Theorem 4.1 in .
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 havewhere . Obviously, we can further get by condition the following:ThenLet , , and by condition (5), we know that and satisfy the conditions in Lemma 1. Hence, one can obtainwhich implies thatAccording to the condition , we can getwhich 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 andhold, 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 havewhere is called the initial value on , andAccording to expression (14), for any , one can obtainwhere is defined in the same way as in Theorem 1. We refer to the idea in  and set , and , then , , and . From (31), it follows Noting that for all , we can getHence, we obtain the following result:By the generalized Grönwall inequality in Lemma 1, we havewhich implies that, for any ,From (30) and (36), we can conclude thatBy 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 havewhere is called the initial value, andTherefore, 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 haveUsing the same approach in (36), one can getBy (42) and (44), we can obtainTherefore, 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 havewhich implies that the fractional system (1.4) has finite-time stability on , by the Definition 5.
As , , we haveTherefore, 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.
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 haveTo 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 obtainAssume that , . It is obvious thatLet , , and . By calculating, we can choose . By Mathematica software, we haveand 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.
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.
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.
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).
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