New Fixed-Time Stability Theorems for Delayed Fractional-order Systems and Applications

This paper examines the problem of improved fixed-time stability for generalized delayed fractional-order systems (FOSs). As a first step, some stability conditions are presented in two theorems to verify fixed-time stability (F-TS) of FOSs by using Lyapunov stability theory and indefinite Lyapunov functionals, where the fractional-order derivative of the indefinite functions may not exist. Furthermore, the corresponding estimated settling time of FOSs is also provided. Second, the results are extended to study fractional-order neural networks (FONNs) with time-delays in fixed-time synchronization. Using the Dirac delta functions, we propose an explicit saturated impulsive controller to synchronize the master and slave systems. Moreover, by constructing suitable indefinite Lyapunov-Krasovskii functions (LKF), we derive algebraic conditions to guarantee the fixed-time synchronization of the addressed FONNs. The simulation results demonstrate the feasibility and efficacy of the proposed method.


I. INTRODUCTION
The literature indicates that arbitrary order calculus (or fractional calculus) appeared around the same time as classical calculus. Fractional calculus did not attract attention until recent years when scholars found that classical calculus cannot explain many random errors, whereas fractional calculus will be able to describe some unusual diffusion processes. Because of its memory and hereditary properties, fractional calculus has become widely used in a variety of fields. The flow of blood, electrolysis, viscosity, and similar phenomena are examples of these phenomena [1]- [3]. Many promising results have been published in the past few years, particularly in the area of stability analysis of FOSs. Yang et al. [4] study impulsive fractional-order nonlinear systems with quadratic Lyapunov functions. In their paper, Liu et al [5] apply the linear matrix inequality method to the case of uncertain FOSs to extend the results obtained by Yang et al [4].
The neural networks (NNs) are computational models that contain interconnected nodes that perform functions similar to the brain's neurons. In particular, they are useful for signal processing, pattern classification, convex classification, cognitive control, etc. [6]- [8]. Note that NN internal dynamics, such as stability, multistability, synchronization, etc. [9]- [13], each of which has become increasingly important in recent decades. FONNs have therefore been investigated in recent years to determine their stability over a finite time period, see [14]- [16]. FONNs, which are a type of fractionalorder dynamical system, have recently been discussed extensively. Since FOSs contain memory and heredity, they have an advantage in information processing, parameter estimations, and a variety of other artificial intelligence tasks. Many great and fascinating discoveries have been made using FONNs in the areas of stability [17], stabilization [18], and synchronization [19].
It is inevitable that there will be delays in many electronic networks. Due to the finite switching speed of amplifiers and the inherent time required for information transmission between neurons, the time-delay produces nonlinear dynamics such as oscillation, bifurcation, chaos, and even instability [20]. A delay in the response of a neuron might affect the network's stability, resulting in oscillatory and unstable characteristics. Therefore, it is necessary to study the networks stability. Thus FONNs with various stability has been studied [21]- [25], such as global stability, Mittag-Leffler stability, finite-time stability (FN-TS) and asymptotic stability. Additionally, a great deal of attention has been paid to synchronization as a major and intriguing phenomenon of NNs by a large number of scientists from a variety of different domains. Synchronization means that two or more systems share a common dynamical behavior by coupling or external force. Because of its wide application in the fields of secure communication, biological system, information processing and so on, synchronization control has been extensively studied and many good results have been achieved [27]- [30]. Among them, the global stability analysis of FONNs is investigated in [21]. By utilizing the geometrical properties of activation functions and algebraic properties of nonsingular M-matrix, the coexistance and multistability of multiple equailibrium points of FONNs are obtained in [22]. By using the generalized reciprocally convex inequality and novel LKfs, several stability criteria for the considered NNs are investigated in [26]. Based on the maximum modulus principle and the spectral radii of matrices, [29] studies the delay independent stability criteria for the FONNs. Some results for FONNs with unbounded time-varying delays are derived to ensure that the equilibrium points of the nonlinear system is asymptotically stable in [30].
Synchronization can be divided into two broad categories based on synchronization time: asymptotic synchronization and finite-time synchronization (FN-TSY). As time passes, driving and response systems achieve asymptotic synchronization. There are several types of synchronization, including exponential synchronization, hybrid synchronization, lag synchronization, and cluster synchronization. The practice of synchronization in indefinite time is often unrealistic, we consider only one example: in secure communication, the longer the synchronization period, the greater the probability that the information will be broken, and that more information will not be successfully received during transmission.
Additionally, in some engineering domains, it is always expected that the synchronization will be completed within a short period of time, which is commonly referred to as FN-TSY. While asymptotic synchronization is better when it comes to robustness and anti-interference properties, FN-TSY is also better in terms of synchronization time since it achieves optimality. However, one disadvantage of FN-TSY is that its settling time is dependent on the initial condition of the system. The fact that many engineering systems' initial values are no longer available may result in significant inconvenience for their practical implementation. A solution to this problem has been established by [31], where the settling time is restricted at its upper limit by a fixed number that is independent of the initial value. Calculating the settling time is a major challenge in F-TSY, as it depends on the initial values of the original system. Generally, different initial values will result in different convergence times and different forecasts. From the perspective of the application, it is obvious that it is problematic. Therefore, it is necessary to learn about the convergence times of solutions with initial values. Moreover, many practical systems, such as power systems and spacecraft dynamics, also require FN-TS for some realistic reasons and needs. It is therefore important to examine both the theory and the applications of F-TS of dynamic systems.
Lyapunov function plays an important role in analysing systems stability, because it does not require the explicit solution of the corresponding ordinary difference or differential equations. By this method, the asymptotic stability is guaranteed if the time-derivative of the Lyapunov function (a positive definite function) along the solutions is negative definite [32]. When the time-derivative of the Lyapunov function is negative semi-definite, stability rather than asymptotic stability follows [33]. Thus, construction of Lyapunov functions has been a fascinating subject. However, the results of previous work [34]- [37] have a common requirement that the time derivatives of Lyapunov functions of systems (or subsystems) must be negative in the convergence process. To relax this restriction, Chen et al [38] gave an improved result that the time derivatives of Lyapunov functions are allowed to be indefinite. To relax these restrictions, we will use an analysis tool named indefinite Lyapunov function, which is introduced in the work of Ning et al [40] and used to analyze the input-to-state stability of nonlinear timevarying systems. In recent decades, indefinite derivative LFs have been proposed as a framework for investigating the synchronization of various types of dynamical systems, such as nonlinear systems [39]- [41], switched nonlinear time varying systems [42], [43], inertial NNs [44], etc., illustrates how indefinite Lyapunov derivatives can be used to study the F-TS of nonlinear time-varying systems with switched perturbations, where positive definite Lyapunov functions have an indefinite derivative to study the F-TS property. Comparing with the traditional Lyapunov function, this tool allows the time derivatives of Lyapunov functions of subsystems to be indefinite and have a tighter upper bound. This approach has the advantage that, as we have already stated, the derivative of the Lyapunov function need not be negative or negative semi-definite.
Another critical concern is the synchronization control of FONNs to the desired trajectory, which has been wellformulated as derive-response synchronization, which is accomplished using linear feedback control [45], and active control [46]. Impulsive control generally has lower control costs, greater confidentiality, and greater resilience than other types of control. It has been widely applied in a variety of fields, including financial markets, static multisynchronization, stability, and information security. Accordingly, the majority of existing results on the design of impulsive controllers [4], [23] are based on the assumption that impulsive strength is not restricted. In order to achieve the desired control performance, we can artificially increase the impulsive strength of the system. From an applications perspective, it is actually very difficult to accomplish the design aim for every control input since actuator saturation is ubiquitous in practically all control systems, making it difficult to achieve the design objective for every control input. When a car travels over speed bumps at a high rate of speed, a reduction in speed may be experienced. Speed bumps serve as impulses, and the reduction in speed may lead to saturation. As a consequence, the saturation of impulses can have a significant impact on the dynamics of NNs. If such restrictions are not managed carefully, or even if the appropriate controllers fail to take them into consideration, then detrimental behaviors may result. Therefore, when investigating the impulsive control of FONNs, it is essential to consider the influence of saturated impulses.
Based on the above discussion, the primary goal of this paper is to study the novel F-TS theorems for delayed FOS and to obtain a more accurate settling time as a result of this study. Second, using the newly discovered F-TS theorems, the F-TSY of the zero solution of a class of FONNs with time-delays is investigated, and some new F-TSY results are derived. Among the contributions of the paper are: 1) In light of the preliminary results of fractional calculus, some new results are imposed on the indefinite LKF and the framework of a Caputo fractional-order derivative introduced in [47], which leads to the new F-TS theorems. The well-known results presented in [48] are significantly improved over the previously published results.
2) The newly established new F-TS theorems can be applied to achieve the F-TSY of delayed FONNs with time-delays by incorporating the newly established F-TS theorems. For delayed FONNs, we propose an explicit saturated impulsive controller based on delta functions. The developed saturated impulsive FOSs reveal a significant result that impulsive control has an influence on controlled FOSs that is both dependent on impulsive function and related to the FOSs' order.
Certain prior conclusions about the FN-TS and F-TS of FONNs can be extended.
The rest of this paper is organized as follows: Section II explains the FOS architecture and provides some lemmas. In Section III, the F-TS theorems for considered FOSs with an indefinite LKF approach are presented. In section IV, the saturated impulsive controller is used to analyze the F-TSY of FONNs. The effectiveness and correctness of the presented results are demonstrated by some simulations, and the conclusion of this paper is briefly discussed in Section V and Section VI, respectively.

II. PRELIMINARIES AND MATHEMATICAL MODEL DESCRIPTION
There are some fundamental definitions of fractional calculus in this section, as well as some lemmas that are required during the demonstration of the theory and problem statement.

A. FRACTIONAL CALCULUS BACKGROUND:
A fractional derivative can be defined in a number of ways, but three commonly used definitions are Grunwald-Letnikov, Riemann-Liouville, and Caputo (see [47]). In general, these three definitions are not equivalent. Definition 1: [47] The Caputo fractional-order derivative of order q of a continuous function w(x) is defined as follows: where 0 < q < 1, the term D q stands for the Caputo fractional-order derivative and the Gamma function is defined in the same manner as before Definition 2: [47] For a Lebesgue-integrable function w(x), the fractional-order integral of order q is defined as follows: The Caputo definition is the one that is most frequently encountered in Physical-Chemistry applications, out of the three definitions listed above. Because it simply requires initial conditions in terms of integer-order derivatives, the Caputo derivative is more applicable to real-world problems because it is easier to understand. Lemma 1: [49] For q ∈ (0, 1), p ∈ R, there holds Lemma 2: [47] Suppose that the Caputo fractional-order derivative D α w(x) is integrable, then Particularly, for 0 < q < 1, one has

Remark 1:
The fractional-order operator D q is a nonlocal operator since its fractional-order derivative depends on the integral's lower boundary. In contrast, the derivative of integer order is unquestionably a local operator.

B. MATHEMATICAL MODEL DESCRIPTION
Consider the following fractional-order differential system with time-delay: Here, w(t) = (w 1 (t), w 2 (t), . . . , w n (t)) T ∈ R n is the state This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3183149 w n (t − σ)) T is the delayed state variable; σ is a constant time-delay; let g : R × R n × R n → R n be Lebesgue measurable and essentially locally bounded; ; γ = sup θ∈[−σ,0] γ(θ) . Definition 3: The origin of (1) is said to achieve fixed-time (F-T) stable, if it is finite-time (FN-T) stable and there is a constant T > 0 such that the settling time T * (t 0 , w 0 ) ≤ t 0 + T , for any initial condition w 0 ∈ R n .
It is important to note that a C-regular Lyapunov function V (w) is not always differentiable in the usual sense. Our goal in this article is to establish new criteria for the F-TSY of delayed fractional-order differential systems (FDSs) based on indefinite functionals, which will be discussed further below.

III. FIXED-TIME STABILITY/SYNCHRONIZATION THEOREM'S
This section presents some novel F-TS or synchronization theorems for the delayed FDS (1). Theorem 1: Suppose there exists a C−regular function V (w(t)) : R n → R, for any indefinite function A (t) and non-positive function B(t) such that any solution w(t) of (1) satisfies the inequality where 0 < α < β < q, and A (t), B(t) satisfy the following inequalities , and , 1 , 2 , δ 1 , δ 2 are positive constants. Then V (w(t)) satisfies the following inequality for t 0 ≤ t ≤ T * and V (w(t)) = 0 for all t ≥ T * . Then the zero solution of the FDS (1) is F-T stable and the corresponding settling-time is and By the condition (2) and , If inequality (2) is true, then a nonnegative function C (t) exists such that From Lemma 1, it follows that Using Lemma 2, we can integrate (8) from 0 to T 0 and get We can deduce from fractional integral Definition 2 and (t − θ) −(q−1) > 0, Γ(q) > 0, ∀θ > 0 C (θ) ≥ 0, then we can easily obtained that This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. In addition, and From (9)-(12), one has .
Then, a nonnegative function C (t) exists such that Multiplying (14) by Γ(1+q−α) Γ(1−α) V −α (w(t)) on both sides and using Lemma 1 we can get Using Lemma 2, we can integrate (15) from t 0 to t and get similar to (10)-(12), we can obtain Therefore, we get V (w(t)) = 0 for all The proof of the Theorem is completed. 2 Corollary 1: Suppose there exists a C−regular function V (w(t)) : R n → R, for any non-positive function B(t) such that any solution w(t) of (1) satisfies the inequality where 0 < β < q and B(t) satisfies the following inequality where 2 , δ 2 are positive constants. Then the zero solution of the FDS (1) is F-T stable and the corresponding settling-time is calculated by: Based on the proof of Theorem 1, the conclusion is obvious. Remark 2: Evidently, Theorem 1 presents a novel result for delayed FDS's F-TS, which differs from earlier results [48]. When A (t) ≡ 0 and B(t) = −a, where 'a' is a nonnegative constant, the inequality (2) will reduce to the inequality proposed in [48] as following: Therefore, Theorem 1 includes and extends the results of [48]. Theorem 1 is more generalized than the Lemma proposed in [48]. Hence, the F-TS/F-TSY of the FDS is extended. Remark 3: In particular, the settling-time of Theorem 1 could be mathematically calculated by the following equality from the above equality, the settling-time T * depending on the parameters β, α, , 2 , 1 , δ 1 , δ 2 and fractional-order q. Theorem 2: For the FDS (1), if there exists a C−regular functionV (w(t)) : R n → R, two functions A (t), B (t) such that the following inequality is hold: 6 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. whereᾱ,β, κ > 0 and satisfying 0 <ᾱκ <βκ < q and A (t), B (t) satisfies the following inequalities where 1 , 2 , δ 1 , δ 2 are positive constants. Then the zero solution of the FDS (1) is F-T stable and the corresponding settling-time is estimated by: Proof 2: To begin, suppose there is a t 0 such that V (w (t 0 )) ≤ 1, and If not, suppose thatV (w(t)) > 1, ∀t ∈ [0, T 0 ], then which implies thatV (w (T 0 )) > 1 and we also haveVᾱ(w(t)) ≤ Vβ(w(t)), 0 <ᾱκ <βκ . By the condition (17), we can have, For a nonnegative function S (t) and multiplying by Γ(1+q−βκ ) Γ(1−βκ )V −βκ (w(t)) on both sides above formula gives that and from Lemma 1 Using Lemma 2, we can integrate (21) from 0 to T 0 and get we can deduce from fractional integral Definition 2 and (t − θ) −(q−1) > 0, Γ(q) > 0, ∀θ > 0 S (θ) ≥ 0 and similar to (10)-(12), we can obtain thus, we haveV (w(T 0 )) < 1, which is a contradiction tô V (w (T 0 )) > 1. On the other hand, κ then a nonnegative function S (t) exists and similar to above process we can integrate from t 0 to t, one haŝ therefore, we getV (w(t)) = 0 for all.
The proof of Theorem 2 is completed. 2 Corollary 2: For the FDS (1), if there exists a C−regular functionV (w(t)) : R n → R, and function B (t) such that the following inequality is hold: whereβ, κ > 0 and satisfying 0 <βκ < q and B (t) satisfies the following inequality Proof 3: Based on the proof of Theorem 2, the conclusion is obvious. Remark 4: Previous publications have extensively utilized Lyapunov function theory to analyze the stability and synchronization of FONNs and delayed FONNs. Their conducted Lyapunov functions, however, have all negativedefinite derivatives. The fractional derivative of function V (w(t)) in Theorem 1 is indefinite, that is, it can be unbounded, negative definite, or positive definite. Accordingly, Theorem 1 has a better chance of achieving the F-TS/F-TSY for delayed FONNs. It is possible to improve previous findings on FN-T and F-TS/F-TSY of FONNs.

IV. FIXED-TIME SYNCHRONIZATION OF DELAYED FRACTIONAL-ORDER NEURAL NETWORKS
FDSs have received considerable attention to date because they have a number of practical applications. Few studies have been conducted on the dynamics of FONNs with saturated impulsive control schemes. Up until now, only a few investigations have been conducted on the F-TSY theorem for FONNs. Moreover, many earlier studies of the stability or synchronization of FONNs were based on the negative definite of the derivative of the Lyapunov functional. Therefore, the F-TSY of FONNs with saturated impulsive control must be studied in more depth, and the derivative of the Lyapunov functionals must be indefinite. In this section, we consider the F-TSY of delayed FONNs based on Theorems 1 and 2.

A. SYSTEM DESCRIPTION
The dynamics of the pth neuron of FONNs with time-delay is represented by n r=1 a pr g r (w r (t)) + n r=1 b pr q r (w r (t − τ r )) + I p , where p = 1, 2, . . . , n, d p > 0, indicate the rates of neuron self inhibition in a pth neuron; w p (t) denote the state variable; g r (·) is the activation function; I p represent external vector; τ r correspond to the time-delay and satisfies 0 ≤ τ r ≤ τ, where τ is a constant. Let w(t) = (w 1 (t), w 2 (t), . . . , w n (t)) T . System (23) is referred to as the driving system, while following system (24) is the associated response system where ν p (t) is controller input to be designed later. Let w(t) = (w 1 (t),w 2 (t), . . . ,w n (t)) T .
The appropriate controller ν p (t) is designed in the following way: where h ∈ N + , S p (t) = K p e p (t), e p (t) =w p (t) − w p (t) is the synchronization error. K p is the impulsive control gain to be determined. A (t) is an indefinite function, B(t) is a negative function and 0 < α < β < q. δ(·) is the delta function with the time sequence {t h , h ∈ N + }, it is a strictly increasing sequence of impulsive instances and assumed that lim h→∞ t h = +∞, e p (t h ) = e p t − h = lim t→t − h e(t). However, in practice, controllers can only deliver a certain amount of signal due to physical or safety limitations, resulting in an inability to attain the required results. This issue can best be addressed by implementing an impulsive controller with actuator saturation: , sat (S 2 (t)) , . . . , sat (S n (t))] T ∈ R n is the saturation function, sat (S p (t)) = sgn (S p (t)) min {∆, |S p (t)|} where S p (t) ∈ R and ∆ > 0 is the known saturation level. Then the controller ν p (t) is designed as follows: The proposed controller (26) represents the synchronization error systems as −a pr g r (w r (t))} + n r=1 {b pr g r (w r (t − τ r )) The initial conditions of above synchronization error system (27) is e p (θ) =γ(θ) − γ(θ).
To obtain the F-TSY criteria for the delayed FONNs (27), the following Assumptions are given 8 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and Assumption 1: Assume that for any positive constant G r that meets the following condition: Assumption 2: The controller is perturbed by the control parametric uncertainty ∆K p , which meets the following condition: Define a parameter that changes over time u p (t h ) as Without a doubt, it possesses u p (t) ∈ (0, 1], and the saturation input can be stated as The synchronization error system (27) can then be described as follows: Lemma 3: For any function e(t) ∈ C 1 [(0, +∞), R] and 0 < q < 1, then D q |e(t)| ≤ sign(e(t))D q e(t), t ≥ t 0 holds almost everywhere. In the next theorem, a new F-TSY criteria will be developed for the delayed FONNs (28).
(1 + ρψ (t h )) KU (t h ) + I n ≤φI n (31) then the response system (24) is synchronized with derive system (23) in a F-T via the following controller where 0 < α < β < q, and A (t), B(t) satisfies the inequalities 1 where , 1 , 2 , δ 1 , δ 2 are positive constants. Then the settling time is estimated as Proof: Consider the following Lyapunov functional candidate: Based on Lemma 3, the Caputo fractional-order derivative of V (e(t)) can be determined for t = t h by deriving the above expression as sgn (e p (t)) D q e p (t) from Assumption 1, we have that

VOLUME 4, 2016
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. When t 1 < t ≤ t 2 , similarly, we have V q−α (e (t)) ≤φ q−α V q−α (e(t + 0 )) ≤V q−α (e(t + 0 )) , we can obtain that V q−α (e (t)) ≤V q−α (e(t + 0 )) In general, for any t h < t ≤ t h+1 , we can obtain that ≤φ hq−hα V q−α (e(t + 0 )) therefore for any t ∈ (t h , t h+1 ], we can have V q−α (e (t)) ≤V q−α (e(t 0 )) It follows from Theorem 1 that the derive and response delayed FONNs (23) and (24) can achieve the F-TSY under the controller (32) and the settling time can be estimated by This completes the proof.
In [50], the bipartite fixed-time synchronization problem for fractional-order signed neural networks with discontinuous activation is discussed. In which the Filippove multimap is used to convert the F-TS of the fractional-order general solution into the zero solution of the fractional-order 12 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and differential inclusions. In our work [50], we state and illustrate the F-TS lemmas of the delayed discontinuous systems, where the results are derived by using the second meanvalue theorem for definite integrals [51] for the fractionalorder 0 < ρ < 1. In the present paper, we considers the improved fixed-time stability problem of generalized delayed FOSs by using well known Lemma 1 in [49] and the concept of Gamma functions. In both research results are imposed on the indefinite LKF and the framework of a Caputo fractionalorder derivative introduced in [47]. Comparing with the results of [50], in this paper we propose an explicit saturated impulsive controller based on delta functions. The developed saturated impulsive FOSs reveal a significant result that impulsive control has an influence on controlled FOSs that is both dependent on impulsive function and related to the FOSs' order. The fractional-order delayed discontinuous system is proposed in in [50]. Consider an time-delayed 2ndimensional FONNs (43) with the following discontinuous activations: Figure 5, depicts the results of the numerical simulation of state trajectories of (43) and (44) with the discontinuous activation (46). Figure 6, demonstrates the state responses of FONNs (23) and (43) with different fractional-orders. When fractional-order q is increases then convergence between master and slave system is suddenly occurs see Figure 6(d) and 6(f). But in case fractional-order q decreases the master- slave systems are not converges exactly see Figure 6(b). This shows that the fractional-order parameter q play an important role in the problem of F-TSY. Example 2: For p = 1, 2, 3, we investigate the following time-delayed fractional-order drive NNs: a pr g r (w r (t)) + n r=1 b pr q r (w r (t − τ r )) + I p (47) correspondingly, and the response system is given by the fractional-order differential equations shown below: with the controllers being designed in the following ways: Let g 1 (w(t)) = g 2 (w(t)) = g 3 (w(t)) = tanh(w(t)), g 1 (w(t − τ )) = g 2 (w(t − τ )) = g 3 (w(t−τ )) = tanh(w(t−τ )), τ = 0.01. Thus, Assumption 1 is satisfied with G 1 = 0.5, G 2 = 0.5, G 3 = 0.5. Let (48) be the response systems and the corresponding derive system is defined in (47). From (47) and (48), we can design an fractional-order error system in such a way that tanh(e p (t − τ r )) + ν p (t), If ν p (t) = 0, Figure 5 depicts the results of a numerical simulation of state trajectories of (47) and (48) with initial values w 1 (θ) = 0.2,w 2 (θ) = −0.4,w 3 (θ) = 0.02, θ ∈ [−1, 0], in two and three-dimensional state spaces. Now let us take the control gain parameters q = 0.9, α = 0.2, β = 0.4, η = 1,φ ∈ (0, 1), ρ = 0.6, A (t) = − t 2 | cos(t)| + 1 1+t 2 , B(t) = − 1 t . Since the conducted Lyapunov function becomes more generalized as a result of the fact that the derivative of it is indefinite almost everywhere. It has the potential to significantly improve the standard Lyapunov function, which is negative and is used to determine the stability results of FONNs. By the direct computations, we can easily verify the conditions (29)-(34) holds with the corresponding impulsive controller gain determined as K = 0.499I 3 and the discontinuous controller inputs ν p (t) can be designed as bellow: ν p (t) = − sgn (e p (t)) η − t 2 | cos(t)| + 1 1 + t 2 |e p (t)| sat (S p (t)) δ (t − t h ) , Therefore, we may conclude that the fractional-order driveresponse system (47)   dance with Theorem 3. This is illustrated in Figures 6-7 with the initial values (0.2, -0.4, 0.02). If we take the impulsive gain K = 0.99I 3 , then it does not converges, it goes against the control protocol which is shown in Figure 8. Figures 9 and  10 shows when other parameters are fixed, we have that the relationship between the settling time and the settling time parameters q, α, β, , 1 and 2 .

VI. CONCLUSION
In this paper, we investigated improved F-TS and F-TSY problems for delayed FOSs. We have obtained F-TS theorems for delayed FOSs with indefinite functionals. The fixed-stability theorems for FOSs using indefinite Lyapunov functionals have been derived using a fractional-order Caputo derivative operator. Our goal is to determine whether F-TS theorems hold for FONNs with time-delay. Based on a delta function for delayed FONNs, we have developed an explicit saturated impulsive controller. Numerical examples illustrate our theoretical findings.
Our next research will be focus on how to design periodically intermittent control and sampled data control to realize finite-and fixed-time synchronization criterion for delayed FONNs.