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1 Introduction

Impulsive control has wide applications in real world. Some useful impulsive control approaches have been presented in many fields such as in financial models, epidemic models, neural networks and so on [6, 7, 17, 19, 21, 25]. As is known to us, impulsive control is a discontinuous control. In some situation, it can perform better than continuous case for special control purpose. There has been great interest in this area as witnessed by scholars new contributions. Compared with instantaneous impulses, the action of non- instantaneous impulses still starts at an arbitrary fixed point but it remains active on a finite time interval. While, there are few works about impulsive control concerning noninstantaneous impulses.

Coupled systems of differential equations on networks (CSDENs) have been widely applied in various fields of biology, engineering, social science and physical science such as in modeling the spread of infectious diseases in heterogeneous populations, neural networks, ecosystems and so on [4, 5, 12, 24, 28]. Especially, the stability analysis of CSDENs is one of the most essential topics in practice. Li and Shuai [16] proposed a new method by combining graph theory with Lyapunov methods to investigate global stability problem for CSDENs. Since then, Suo [23] applied results from graph theory to construct global Lyapunov functions and then established a new asymptotic stability and exponential stability principles. However, many results about stability of coupled system on networks utilize differential equations of integer order [22, 26]. Until now, there are few relevant researches about stability analysis for coupled systems of fractional-order differential equations on networks (CSFDENs). Li and Jiang [14] investigated CSFDENs, they obtained a global Mittag-Leffler stability principles by Lyapunov method and graph theory. Recently, Li [15] studied stability of fractional-order impulsive coupled nonau- tonomous (FOIC) systems on networks using graph theory and Lyapunov method to get stability for a kind of FOIC systems. Some remarkable achievements have been made in [8, 13–15, 18, 22, 26] during the past few years.

Practical stability analysis is one of the most important types for stability theory. In 2016, Stamova [20] derived the practical stability criteria of fractional-order impulsive control systems by using fractional comparison principle, scalar and vector Lyapunov- like functions. In 2017, Agarwal [2] investigated practical stability of nonlinear fractional differential equations with noninstantaneous impulses and presented a new definition of the derivative of a Lyapunov-like function; see literatures [2, 3, 9, 11, 20] for more details. The purpose of this paper is to study the practical stability for a class of impulsive CSFDENs with noninstantaneous impulses. Generally speaking, we investigate systems on networks by studying each individual vertex dynamics to determine the collective dynamics and explore the noninstantaneous impulses effect on systems. We establish new practical stability criteria for the systems. Some sufficient conditions are given to meet the practical stability, uniform practical stability and practical asymptotic stability of this coupled systems on networks.

Our results generalize relevant results in [2]. We provide a new method of impulsive control law for impulsive CSFDENs with noninstantaneous impulses by using graph theory and Lyapunov method. The systems in [2] can be considered as a special case for It is the first time to consider fractional-order coupled systems with nonin- stantaneous impulses via graph theory. We also illustrate that the topology property of systems have a close connection with the practical stability of the systems.

Compared with the existing method for studying impulsive CSFDENs, we develop a systematic approach to construct a Lyapunov-like function by using the Lyapunov- like function of each vertex system, which avoids the difficulty of finding it directly of the whole system. Especially for systems with noninstantaneous systems, it is a creative work. In this paper, we are interested in whether the dynamical behaviors can be effected by network encoded in the directed graph. Therefore, to better solve this problem, we construct piecewise continuous Lyapunov-like functions in each vertex system, then construct a global Lyapunov-like function for coupled systems as Besides, this method constructs a relation between the practical stability criteria and topology property of the network, which can help analyzing the practical stability of fractional-order complex networks.

The rest of our paper are organized as follows. In Section 2, we introduce some necessary notions, definitions and lemmas. Practical stability criteria about fractional- order coupled systems on networks are given in Section 3. In Section 4, examples are given to show the applicability of our results.

2 Preliminaries

In this section, we recall some basic and essential definitions of fractional calculus as well as concepts and lemmas of graph theories for better obtaining our main results.

The following knowledge of graph theories can be found in [16].

A nonempty directed graph is defined with a vertex se and an edge set , each element of denotes an arc leading from the initial vertex to terminal vertex . Two diagraph , and are given. If then is called a subgraph of . A subgraph of is a spanning subgraph if contains all vertices from .

A digraph is weighted if a positive weight is assigned to each arc. Denote if and only if there exists arc from vertex to in, otherwise,. The weight of denotes the product of the weights on all its arcs. A directed path is a subgraph of with vertices and a set of arcs. If , then is a directed cycle.

Assume that is a weighted diagraph with vertices. is a matrix , whose element equals the weight of each arc . Denote weighted diagraph with weight matrix as. is said to be balanced if covers all directed cycle in means the reverse of constructed by reversing direction of all arcs in . A connected subgraph is a tree if it has no cycle. We call the root of a tree if is not a terminal vertex of any arc and each of the remaining vertices is a terminal arc of one arc. A subgraph is a unicyclic graph when it is a disjoint union of root trees, whose roots form a directed cycle. and are unicyclic graphs with the cycles and respectively. When is balanced, The Laplacian matrix if is defined as for for The constant denotes the maximum eigenvalue of matrix .

Lemma 1. (See [16].) Assume Let denote the cofactor of the ith diagonal element of L. Then the following equation holds:

where are arbitrary functions, are the elements of L, is the set of all spanning unicyclic graphs of is the weight of denotes the directed cycle of .

If is balanced, then

and if is strongly connected, then

Definition 1. (See [27].) The Riemann–Liouville fractional integral of order of a function is given by

where is the Gamma function, provided the right side is pointwise defined on

Definition 2. (See [27].) The Caputo fractional derivative of order of a function is given by

where is the smallest integer greater than or equal to, provided that the right side is pointwise defined on

In case we have

Definition 3. (See [2].) We say if is differentiable, the Caputo derivative exists and satisfies (1) for

Now, we introduce the definition of Grunwald–Letnikov fractional derivative and Grunwald–Letnikov fractional Dini derivative, then we use the relation between Caputo fractional derivative and Grunwald–Letnikov fractional derivative to define Caputo fractional Dini derivative. The details can be found in [10].

Definition 4. (See [10].) The Grunwald–Letnikov fractional derivative of a function is given by

and the Grunwald–Letnikov fractional Dini derivative of a function is defined as

whereare the Binomial coefficients, and denotes the integer part of

Definition 5. (See [10].) The Caputo fractional Dini derivative of a function is defined as

Consider a network represented by a diagraph with vertices. A fractional-order impulsive control coupled system with noninstantaneous impulses can be built on by assigning dynamics on each vertex, then coupling these individual vertex dynamics in . In this way, for the vertex dynamics is defined as the following system:

where are two increasing sequences such that be a given arbitrary point. Without loss of generality, we make an assumption that

The solution of the vertex system (2) satisfies

We can refer to [2] for detailed proof.

Let be a given interval, for We introduce the following class of functions:

Remark 1. From the above description of any solution for system (2) we can conclude that of (2) is discontinuous at points

Definition 6. (See [2].) Let be a given interval. are given sets.We say that the fuctions belong to the classes if

(i) The functions are continuous onand they are locally Lipschitzian with respect to the second argument;

(ii) For each there exist finite limits

and the following equalities are valid:

For we define the generalized Caputo fractional Dini derivative with respect to system (2) as

where and for any there exists such that

Together with system (2), we consider the scalar comparison system on graph. The vertex dynamics is described as follows:

where

Next, we prove some comparison results for noninstantaneous impulsive Caputo frac- tional-order system (2) using Definition 4 for fractional Dini derivative. Without loss of generality, we assume We will use results in Lemma 2 of [2] to obtain comparison results for system (2).

Lemma 2. (See [2].) For we let:

(i) The function is the solution of initial value problems (IVPs) for the ith vertex system (2), where

(ii) For he functions are such that

(iii) The function and for a given the IVPs for the ith vertex of the scalar system (3) has a maximal solution

(iv) The function and the following inequalities hold:

Then the inequalityimplies on

Corollary 1. For if the function satisfies

Lemma 3. For if the function satisfies

Proof. Take in system (3). The solutions of IVPs for (3) satisfy

Without loss of generality, we assume LetWe prove by induction.

Let be a solution of (2).

Then

the conclusion follows from Corollary 2.3.2 in [1], i.e., inequality (4) holds on

Let Then we get

After that, still by Corollary 2.3.2 in [1] we have

where So inequality (4) holds on We can continue this process, then induction proves that Lemma 3 is true.

Remark 2. The results of Lemmas 2, 3 and Corollary 1 are true on the half-line. In other words, conditions in Lemma 2 are satisfied for Then the conclusions still hold.

3 Practical stability analysis for fractional-order coupled systems with noninstantaneous impulses on networks

In this section, we investigate the following fractional-order coupled system with nonin- stantaneous impulses on graph :

for whererepresents the influence of vertex on vertex . If there is no arc from to in graph ,

The following assumptions are given:

(H1) The function satisfy the globally Lipschitz conditions.

(H2) The functions for satisfies the globally Lipschitz conditions.

If (H1)–(H2) are satisfied, IVPs for (5) has a trivial solution

For we define the generalized Caputo fractional Dini derivative with respect to system (5) as

where and for any there exists such that

Let

We define as follows:

where c. is defined in Lemma 1,

Definition 7. The zero solution of system (5) is said to be

(S1) practically stable with respect to if there exists such that for any the inequalityimplies

(S2) uniformly practically stable with respect to if (S1) holds for every

(S3) practically asymptotically stable with respect to if (S1) holds and

Remark 3. From Definition 7 we can see that practical stability is neither stronger nor weaker than stability in the sense of Lyapunov. Practical stability is not defined in the neighborhood of the origin, but an arbitrary set. To some extent, the range of this set can better reflect the essence of the study of practical problems. In detail, a system considered may be unstable in the sense of Lyapunov stability, whereas in practical problems, the dynamic behavior of the system can meet the actual demand within a certain range. For example, rocket launchers are considered to have unstable navigation trajectory, while the effect of rocket system under oscillation can be accepted, hence it is practical stability. The key point of the creation for practical stability theory is that practical stability and other means of stability are completely independent concepts.

We use the following sets:

1. is strictly increasing and

2.

Theorem 1. For let the following conditions be fulfilled:

(i) The functionis such that

(ii) The function and for a given the IVPs for the scalar system (3) has maximal solutions

(iii)There exist functions and a matrix in which such that

(iv) Along each directed cycle of the weighted digraph is strongly connected,

(v) There exist functions such that

where

(vi) There exist functions such tha

Then the trivial solution of system (5) is practically stable.

Proof. Define a function According to condition (iii), when we get

Making use of Lemma 1 in weighted digraph (G, A), we obtain

Combing condition (iv) with the fact we have

Define

On account of , we can deduce that

Two constants are given, and . Let There exists a such that for

Then from conditions in Theorem 1 and conclusions of (6), by Corollary 1, we have

In view of condition (vi), we derive

Combining (7) and (8), we obtain

for provided that which completes the proof.

Corollary 2. Assume that is balanced such that

Condition (iv) of Theorem 1 is replaced by

Then the trivial solution of system (5) is practically stable.

Remark 4. In Theorem 1, we assume that is strong connected, which means the topology property of coupled system (5) in a close connection with the practical stability of (5). In fact, without the strong connectedness of , we can only judge the practical stability of vertex system, but we can not judge the practical stability of the whole system. We give an example to illustrate.

Given a weighted graph with 3 vertices, where

The Laplacian matrix of is defined as

Through calculation, we get which means that the practical stability of the third vertex can be checked, but the practical stability of the whole system is unable to be determined. So the strong connectedness can definitely have effect on the practical stability.

Theorem 2. Assume conditions of Theorem . hold, and let following condition holds:

(I) There exist functions such that

Then the trivial solution of system (5) is uniformly practically stable with respect to

Proof. For a function where is defined in Lemma 1. Two constants are given such that provided that

Define On account of we can deduce that

If it follows from conditions (v), (vi) and (7) that for

On the other hand, in view of condition (vi), one has

Combining (9) and (10), we obtain for

This proves the uniformly practically stable of the trivial solution of system (5).

Theorem 3. Assume conditions (i).(ii), (iv).(vi) in Theorem ., and let following condi- tion hold:

(iii.)There exist functionsthe matrixwhich such that for

Then the trivial solution of system (5) is practically asymptotically stable.

Proof. Define When in view of condition (iii.), we get

Then by Lemma 3 we can get

We get the fact that the trivial solution of system (5) is practically asymptotically stable. The proof is complete.

Remark 5. Theorems 1–3 provide a technique by graph theory to construct global Lya- punov functions using piecewise continuous Lyapunov functions in each vertex. This method overcomes the difficulty of directly finding appropriate Lyapunov functions. Furthermore, it is easier to obtain the practical stability of these types of frac- tional coupled systems with noninstantaneous on networks.

4 Examples

Example 1. We consider the following fractional-order impulsive control coupled system with noninstantaneous impulses on network:

Here are -dimensional column vectors. The parameters are nonnegative constants, and when are constant matrices. Let

Let be a graph with vertices,

is strongly connected,

Let We now construct Lyapunov-like functions as For through calculation, we have

where Therefore, conditions (i)–(iii) in Theorem 1 are satisfied.

Furthermore, for

So, along each cycle of we have

Thus, condition (iv´) is satisfied.

Also,

where follows that condition (v) in Theorem 1 is satisfied.

At last, let It is easy to verify We can deduce that condition (vi) in Theorem 1 is satisfied.

According to Corollary 1, taking all the factors into consideration, we can conclude the trivial solution of system (11) is practically stable.

Now we give a numerical simulation to illustrate the effectiveness of our results. Let

identity matrix.When

According to Example 1, the above system is practically stable. Numerical simulation can be seen in Fig. 1.

Corollary 3.Let in Example to According to Theorem ., we can conclude that the trivial solution of system (11) is uniformly practically stable.

Example 2. Consider the following fractional-order impulsive control coupled system with noninstantaneous impulses on network:

are -dimensional column vectors. The parametersare nonnegative constants, and when are constant matrix. Let Let be a graph with vertices, is strongly connected,

Let We now construct Lyapunov-like functions as For through calculation, we have

where Therefore, condition (iii´) in Theorem 3 is satisfied.

Furthermore, we can easily get that condition (iv´) is satisfied. In the same way, we can get

where , it follows that condition (v) in Theorem 1 is satisfied.

Then let We can deduce that condition (vi) is satisfied.

According to Theorem 3, we can conclude that the trivial solution of system (12) is practically asymptotically stable.

We give the numerical simulation of to verify the effectiveness of the obtained results. Letidentity matrix. When According to Example 2, the above system is practically asymptotically stable, which can be seen in Fig. 2.

5 Conclusions

In this paper, we investigate a class of fractional impulsive control systems with nonin- stantaneous impulses on networks. We give sufficient conditions to obtain the practical stability, uniform practical stability and practical asymptotic stability of this coupled systems on networks for the first time. Meantime, we provide an appropriate way to construct global Lyapunov-like functions in view of noninstantaneous impulses. Then, using Lyapunov method and graph theory, the practical stability principles are obtained, which have a close relation to the topology property of the networks. Our results general- ize relevant results in [2] to networks and provide an impulsive control law for impulsive control systems with noninstantaneous impulses.

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Notes

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