Practical stability for fractional impulsive control systems with noninstantaneous impulses on networks

. This paper investigates practical stability for a class of fractional-order impulsive control coupled systems with noninstantaneous impulses on networks. Using graph theory and Lyapunov method, new criteria for practical stability, uniform practical stability as well as practical asymptotic stability are established. In this paper, we extend graph theory to fractional-order system via piecewise Lyapunov-like functions in each vertex system to construct global Lyapunov-like functions. Our results are generalization of some known results of practical stability in the literature and provide a new method of impulsive control law for impulsive control systems with noninstantaneous impulses. Examples are given to illustrate the effectiveness of our results.


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 noninstantaneous 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 nonautonomous (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 Lyapunovlike 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 i = 1. It is the first time to consider fractional-order coupled systems with noninstantaneous 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 Lyapunovlike 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 V i in each vertex system, then construct a global Lyapunov-like function V for coupled systems as V (x) = c i V i (x), c i 0. 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 fractionalorder coupled systems on networks are given in Section 3. In Section 4, examples are given to show the applicability of our results.

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 G = [V, E] is defined with a vertex set V = {1, 2, . . . , n} and an edge set E, each element of E denotes an arc (i, j) leading from the initial vertex i to terminal vertex j.
A digraph is weighted if a positive weight a ij is assigned to each arc. Denote a ij > 0 if and only if there exists arc from vertex i to j in G, otherwise, a ij = 0. The weight W (G) of G denotes the product of the weights on all its arcs. A directed path P is a subgraph of G with vertices {i 1 , i 2 , . . . , i n } and a set of arcs {i k , i k+1 , k = 1, 2, . . . , n − 1}. If i n = i 1 , then P is a directed cycle.
Assume that G is a weighted diagraph with n vertices. A is a matrix (a ij ) n×n , whose element equals the weight of each arc (i, j). Denote weighted diagraph with weight matrix A as (G, A). (G, A) is said to be balanced if W (C) = W (−C), C covers all directed cycle in G, −C means the reverse of C constructed by reversing direction of all arcs in C. A connected subgraph is a tree if it has no cycle. We call i the root of a tree if i is not a terminal vertex of any arc and each of the remaining vertices is a terminal arc of one arc. A subgraph Q is a unicyclic graph when it is a disjoint union of root trees, whose roots form a directed cycle. Q and Q are unicyclic graphs with the cycles C Q and −C Q , respectively. When (G, A) is balanced, where F ij (x i , x j ) are arbitrary functions, 1 i, j n, a ij are the elements of L, Q is the set of all spanning unicyclic graphs of (G, A), W (Q) is the weight of Q, C Q denotes the directed cycle of Q.
and if (G, A) is strongly connected, then c i > 0 for i = 1, . . . , n.

Definition 2.
(See [27].) The Caputo fractional derivative of order α > 0 of a function f : [t 0 , +∞) → R is given by where n is the smallest integer greater than or equal to α, provided that the right side is pointwise defined on [t 0 , +∞).
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 x is given by (1):102-120, 2022 and the Grunwald-Letnikov fractional Dini derivative of a function x is defined as where q Cr are the Binomial coefficients, and [(t − t 0 )/h] denotes the integer part of (t − t 0 )/h.

Definition 5.
(See [10].) The Caputo fractional Dini derivative of a function x is defined as Consider a network represented by a diagraph G with n vertices. A fractional-order impulsive control coupled system with noninstantaneous impulses can be built on G by assigning dynamics on each vertex, then coupling these individual vertex dynamics in G. In this way, for 1 i n, the ith vertex dynamics is defined as the following system: where 0 < α < 1, ] be a given arbitrary point. Without loss of generality, we make an assumption that t 0 ∈ [s 0 , t 1 ).
The solution We can refer to [2] for detailed proof.
https://www.journals.vu.lt/nonlinear-analysis Let J ⊂ R + be a given interval, for 1 i n, Ω i ⊂ R mi . We introduce the following class of functions: Remark 1. From the above description of any solution for system (2) we can conclude that locally Lipschitzian with respect to the second argument; (ii) For each t k ∈ J and x i ∈ Ω i , there exist finite limits and the following equalities are valid: i n, we define the generalized Caputo fractional Dini derivative with respect to system (2) as where t 0 ∈ J, and for any t ∈ (s k , t k+1 ) ∩ J, k = 0, 1, 2, . . . , there Together with system (2), we consider the scalar comparison system on graph. The ith vertex dynamics is described as follows: where . . , u n0 ). Next, we prove some comparison results for noninstantaneous impulsive Caputo fractional-order system (2) using Definition 4 for fractional Dini derivative. Without loss of generality, we assume t 0 ∈ [s 0 , t 1 ). We will use results in Lemma 2 of [2] to obtain comparison results for system (2). [2].) For 1 i n, we let: and for a given u 0 ∈ R, the IVPs for the ith vertex of the scalar system (3) has a maximal solution u * , Ω i ), and the following inequalities hold: Then the inequality . The solutions of IVPs for (3) satisfy

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 noninstantaneous impulses on graph G: x i (t) = I k t, x i (t k − 0) , t ∈ (t k , s k ],  1, 2, . . . ) represents the influence of vertex j on vertex i. If there is no arc from j to i in graph G, g ij = 0, i, j = 1, 2, . . . , n.
Definition 7. The zero solution of system (5) is said to be (S1) practically stable with respect to (λ, A), 0 < λ < A, if there exists t 0 ∈ ∞ k=0 [s k , t k+1 ) such that for any x 0 ∈ R m , the inequality x 0 < λ implies x(t; t 0 , x 0 ) < A for t t 0 ; (S2) uniformly practically stable with respect to (λ, A) if (S1) holds for every t 0 ∈ ∞ k=0 [s k , t k+1 ); (S3) practically asymptotically stable with respect to (λ, A) if (S1) holds and lim t→∞ x(t, t 0 , x 0 ) = 0. https://www.journals.vu.lt/nonlinear-analysis 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: Theorem 1. For 1 i n, let the following conditions be fulfilled: , and for a given u i0 ∈ R, the IVPs for the scalar system (3) has maximal solutions u * [s k , t k+1 ], t t 0 .
(iv) Along each directed cycle C of the weighted digraph (G, A), (G, A) is strongly connected, Then the trivial solution of system (5) is practically stable.
Making use of Lemma 1 in weighted digraph (G, A), we obtain n i,j=1 Combing condition (iv) with the fact W (Q) > 0, we have On account of b i ∈ K, i = 1, 2, . . . , n, we can deduce that b ∈ K. Two constants (λ, A) are given, and 0 < λ < A.
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 t t 0 , provided that x 0 ∈ S(λ), which completes the proof.
Condition (iv) of Theorem 1 is replaced by https://www.journals.vu.lt/nonlinear-analysis (iv ) Then the trivial solution of system (5) is practically stable.
Remark 4. In Theorem 1, we assume that (G, A) 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 (G, A), 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 (G, A) with 3 vertices, where The Laplacian matrix of (G, A) is defined as Through calculation, we get c 1 = c 2 = 0, c 3 = 15, 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 1 hold, and let following condition holds: (I) There exist functions a i ∈ K such that Then the trivial solution of system (5) is uniformly practically stable with respect to (λ, A).
If x 0 ∈ S(λ), it follows from conditions (v), (vi) and (7) that for t t 0 , On the other hand, in view of condition (vi), one has Combining (9) and (10), we obtain for t t 0 , This proves the uniformly practically stable of the trivial solution of system (5).
Theorem 3. Assume conditions (i)-(ii), (iv)-(vi) in Theorem 1, and let following condition hold: , the matrix A = (a ij ) n×n in which a ij 0 and d i > 0 such that for i = 1, 2, . . . , n, Then the trivial solution of system (5) is practically asymptotically stable.
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 Lyapunov functions using piecewise continuous Lyapunov functions V i , i = 1, . . . , n 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 fractional coupled systems with noninstantaneous on networks.
. . , n. We now construct Lyapunov-like functions as where F ij (X i , X j ) = |x i y j |. Therefore, conditions (i)-(iii) in Theorem 1 are satisfied. Furthermore, for i = j, So, along each cycle C of (G, A), we have Thus, condition (iv ) is satisfied. Also, where t ∈ (t k , s k ], t k t 0 , x i ∈ S i (ρ), it follows that condition (v) in Theorem 1 is satisfied.
At last, let b i (x) = x , i = 1, 2, . . . , n. It is easy to verify b i ∈ K. 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 µ i = 1, i = 1, 2, . . . , n, β ii = 0, β ij = 1/(n − 1) if i > j, β ij = −1/(n − 1) if i < j, C k = I n /2, I n is n × n identity matrix. When i = j, β ij = −β ji , λ max C k = 1/2. According to Example 1, the above system is practically stable. Numerical simulation can be seen in Fig. 1.
In the same way, we can get where t ∈ (t k , s k ], t k t 0 , x i ∈ S i (ρ), it follows that condition (v) in Theorem 1 is satisfied.
Then let b i (x) = x , i = 1, 2, . . . , n. 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. Let β ii = 0, β ij = 1/(n − 1) if i > j, β ij = −1/(n − 1) if i < j, C k = I n /2, I n is n × n identity matrix. When i = j, β ij = −β ji , λ max C k = 1/2. According to Example 2, the above system is practically asymptotically stable, which can be seen in Fig. 2.

Conclusions
In this paper, we investigate a class of fractional impulsive control systems with noninstantaneous 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 generalize relevant results in [2] to networks and provide an impulsive control law for impulsive control systems with noninstantaneous impulses.