Modeling and Stability Analysis for Markov Jump Networked Evolutionary Games

Zhao, Guodong

doi:https://doi.org/10.1155/2018/5124343

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Mathematical Theories and Applications for Nonlinear Control Systems

View this Special Issue

Research Article | Open Access

Volume 2018 | Article ID 5124343 | https://doi.org/10.1155/2018/5124343

Modeling and Stability Analysis for Markov Jump Networked Evolutionary Games

Guodong Zhao¹

Guest Editor: Weihai Zhang

Received03 Sept 2018

Accepted02 Oct 2018

Published19 Nov 2018

Abstract

This paper investigates the algebraic formulation and stability analysis for a class of Markov jump networked evolutionary games by using the semitensor product method and presents a number of new results. Firstly, a proper algorithm is constructed to convert the given networked evolutionary games into an algebraic expression. Secondly, based on the algebraic expression, the stability of the given game is analyzed and an equivalent criterion is given. Finally, an example works out to support the new results.

1. Introduction

The importance of networked evolutionary games (NEGs) [1] has been recognized many scholars recently. Unlike traditional -player game, an NEG consists of a network graph and a basic game. The nodes in the network represent players and the edges in the network represent interaction relationship among players. Every player in the given game only plays with its neighbors. Furthermore, in the given game, players have the specialized strategy updating rules to adjust their own strategy choices. They are only affected by their neighbors. It coincides with an obvious fact that, in many practical economic activities, each individual is only in contract with its trade partners. This distinguishing feature makes NEGs theory be a very appealing research topic in recent years [2].

In the last decades, a very useful mathematical tool, called the semitensor product method (STP) of matrices, has been proposed by [3]. With this tool, [3] proposed a method to transform Boolean networks, Boolean control networks, and mix-valued logical networks into some kinds of difference equations. Then, one could investigate them by using mathematical tools in control theory.

Up to now, this method has been successfully applied to the analysis and control of Boolean networks and mix-valued logical networks, and many essential results have been obtained, such as the calculation of fixed points and cycles and the controllability and observability of Boolean networks; see [3–19] for details. References [17, 20] presented the recent applications for STP method in engineering and finite-valued systems, respectively. See [20–29] for more recent developments about the application of STP method in logical networks.

In recent years, many scholars have attempted to study NEGs via STP method. For many specific NEGs, they transformed the given NEGs into some proper algebraic expressions, logical networks, for example. Based on “the myopic best response adjustment rule”, [30] designed a proper algorithm to construct the structural matrices of the updating laws for every players in the given NEG. Reference [31] gave the description of the NEG and investigated the relationship between the given NEG and the given logical networks. NEGs defined on finite networks were considered by [32], which converted the given NEGs to a kind of logical networks and solved the optimization problem when one player was regarded as a control. Reference [33] investigated evolutionary game theoretic demand-side management and control for a class of networked smart grid. A class of event-triggered control for finite evolutionary networked games was studied by [34]. Reference [16] studied stochastic set stabilization of -person random evolutionary Boolean games and its applications.

It is worth noting that all of the above results are concentrated on NEGs with pure strategy dynamics except [16]. However, the fact that most evolutionary games are related to random dynamics more or less cannot be neglected. The work [35] considered an interesting evolutionary game. In the game, major player is with infinite time horizon and the other players, called minor players, are with finite time horizons. At any time, when some new players enter the evolutionary game, the number of them is a random variable, which has a distribution depending on the number of the players who have entered the game at that moment. This situation is called random entrance, which is modeled as a Markov chain. In random entrance, there exists a long living player which has interaction relationships with the other players and the interactions are existing for a certain period. This situation is very common, University-Student Games [36], for example.

Actually, we can model the above evolutionary games with random entrance as a kind of networked evolutionary defined on finite networks. For every time, the entered players make up a network. The given game evolves on these finite networks. Thus, this paper considers this kind of networked evolutionary games as Markov jump networked evolutionary games (MJNEGs). In this paper, we would give the formal description of MJNEGs and analyze the stability of MJNEGs.

The main works of this paper are as follows: The STP method is firstly applied to the study of MJNEGs. The method to formulate the given MJNEGs as an algebraic expression is proposed. The stability analysis of MJNEGs is presented in this paper, and an equivalent test criterion is given.

Notations: denotes the set of real matrices. denotes the set of nonnegative real matrices. , where is the -th column of the identity matrix . An matrix is called a logical matrix, if and denote briefly by . Define the set of logical matrices as . is the -th column (row) of matrix . is a probabilistic vector, if , , and . denotes the set of dimensional probabilistic vectors. If and , is called a probabilistic matrix. denotes the set of probabilistic matrices. denotes the default matrix product throughout this paper. Please refer to [3] for the definition and properties of STP. Because is a generalization of the ordinary matrix product. We omit “” without confusion and call it “product”. denotes Khatri-Rao product of and .

2. Preliminaries

This section gives some necessary mathematic tools, which will be used in this paper.

Lemma 1 (see [3]). (1)Consider and as two column vectors. Then, , where is the swap matrix. Especially .(2)Define and . Then, one gets .

Lemma 2 (see [31]). Consider and . Define dummy matrices, called “front-maintaining operator" and “rear-maintaining operator”, respectively, as Then , .

Lemma 3 (see [37]). Consider (or ) as a pseudological (or logical) function. Then, there exists a unique matrix (or ), called the structural matrix of , such that where , , and .

In the rest of this section, we present some basic concepts in networked evolutionary games.

Definition 4 (see [31]). A normal game with two players is a fundamental network game (FNG), if the strategy set is and player’s payoff function is .

3. Main Results

This section firstly gives the description of MJNEGs. Then, the method to formulate the given MJNEGs is given. Finally, this section analyzes the stability of the MJNEG based on its corresponding algebraic expression.

3.1. Description of MJNEGs

At first, we give the description of an MJNEG as follows:(a) A set of finite networks : the topological structure of each network is a connected undirected graph , where is the set of nodes in network , there exists interaction between node and node in network is the set of edges, and .(b) An FNG: if , then node and node play the FNG in network with strategies and at time , respectively.(c) Players’ strategy updating rule: in network , the rule can be expressed as where is the strategy of player at time , , and is the neighborhood of player in the network , that is, if and only if , , and . Obviously, and .(d) Network evolve process: represents the discrete-time homogeneous Markov chain taking values in a finite set with a transition probability matrix as where , for , for any , and represents that the MJNEG evolves on network at time .

In network , at each time, node only plays with its neighbors, and its aggregate payoff is the sum of payoffs gained by playing with all its neighbors, i.e., in which is the payoff function of the given FNG and .

In this paper, we adopt the myopic best response adjustment rule [38], that is, every player forecasts that its neighbors will repeat their last-step strategy choice, and the strategy choice at present time is the best response against its neighbors’ strategies choice of the last one. Based on that, one hasWhen player may have more than one best strategies to choose, define a priority for the strategy choice as follows: has priority over , if and only if . Thus, player updates its strategy according to the following expression:

We let the initial state for nodes , which are not activated at that moment in the network ; i.e., holds.

Remark 5. We assign the initial state for the new participators. It implies an obvious fact that every new participator has the same conditions when they enter the evolutionary game.

In addition to the evolution of strategies for players, the Markov chain decides the probability with which network the given MJNEG stays at a specific time .

The aim of this paper is to study the algebraic formulation and stability analysis of the given MJNEG as a -valued switched logical system with a given switching signal as a Markov chain.

3.2. Algebraic Expression of the Given MJNEG

This subsection algebraically formulates the given MJNEG as Markov jump -valued logical system. To achieve it, we can take the following steps: Find the proper structural matrices of the payoff function for every node in each networks. Find the proper structural matrix of the updating law for every node in each networks. Via the obtained structural matrices and the transition probability matrix of Markov chain , we construct the algebraic formulation for the given MJNEGs.

Step , using the vector form of logical variables, we identify , where , “" denotes that the strategy is equivalent to . Then, when holds, by Lemmas 1, 2, and 3, and (5), the payoff function of player in network can be rewritten aswhere is the structural matrix of the FNG’s payoff function and is the structural matrix of , is the strategy of player at time , , and .

In Step , we consider the following two cases.

Case I. If , divide into equal blocks as where is all possible benefits of player with other players’ strategy .
Next, we find the best response of player to make its benefit maximum. For all , let the column index set , such that If there are more than one maximum columns, i.e., , one can choose the unique column index with the help of the priority of strategy choice given in (7).
By Lemma 2, letting , we can construct the algebraic form of the updating law for node in network as

Case II. If , by Lemma 2, we define where . Then, each new player entering the MJNEG would have the initial state .
Thus, by (11) and (12), one gets the MJNEGs evolving on fixed network as follows: where and .

In Step , by the aforementioned analysis, we formulate the given MJNEG as follows: where is the discrete-time Markov chain and .

Based on the aforementioned analysis, the following algorithm is constructed to formulate the MJNEG.

Algorithm 6. This algorithm contains four steps: (1) Calculate the structural matrix, , of the payoff functions of the player in node , when , for each network by (2) For each network , divide the matrices into equal blocks as and for all , find the column index , such that ;(3) Formulate the MJNEG evolving on network under study as where , , , , and .(4) Finally, one has the algebraic formulation as follows: where .

3.3. Stability Analysis

This subsection investigates the stochastic global stability of the given MJNEG as where and denotes the given Markov chain. In an evolutionary game, some strategy profile has specific meaning, Nash equilibrium, for example. This subsection analyzes the globally stability at in stochastic sense. In addition, because of transformation of coordinates, we assume that holds.

In the following, we give the definition for MJNEG of global stability in stochastic sense.

Definition 7. The given MJNEG with algebraic form (19) is said to be globally stable in stochastic sense at , if, for and , holds.

Denote , where represents the Dirac measure ever the set with . Since is independent from , one has Then, we have

Rewrite as , where , and ; one has where and

Thus, via (22), the following result reveals some property for the given MJNEG.

Theorem 8. The given MJNEG with algebraic form (19) is globally stable at in stochastic sense if and only if, and ,

Proof. Sufficient. If, and , (24) holds, then according to that . It is easy to see that .
Necessity. If system (19) is globally stable in stochastic sense at , then using Definition 7, for and initial distribution of , , which yields that . Note that , then one has, for every , .

With the help of the above theorem, we can get the main result in this paper. The following theorem gives an equivalent criterion for the stability of the given NEG.

Theorem 9. The given system with algebraic form (19) is globally stable in stochastic sense at if and only if there exist vectors , such that the following conditions hold: for , and .

Proof. Sufficient. To prove the part of sufficient, we need first prove the fact that each network has a common fixed point as under conditions (25).
Assume that is not a fixed point of network , then , i.e., , for some . Subsequently, by the second equation in (25), one has that for every , . Note that , then there exists at least one integer such that . Therefore, which is contradictory to the third equation in (25). Thus, is a common fixed point of each network, which implies that , for . Therefore, and for in (23). Then, the first equation of (22) can be rewritten as Let , then where It is worth noting that . Thus, from Lemma 1 in [39] system, (28) is a positive system. Rewrite as , where , , then by (25) and denoting , one has and . Therefore, system (28) is stable by Proposition 1 in [39]. By Theorem 8, network (19) is stochastically globally stable.
Necessity. Because system (19) is stochastically globally stable. From Theorem 8, we have (24). For Markov process is ergodic, one get that there exists such that the probability of reaching every mode is positive after time . Thus, for all networks, is a common fixed point. Otherwise, if is not a fixed point of network , then and for . By (22) and (24), one has With in hands, one obtains that , which implies that holds. It is contrary to that the probability of every reaching mode is positive after time . Therefore, for each , and . So, satisfies (28). System (28) is a positive and stable system. Then, by Proposition 1 in [39], there exists a vector , where and , such that . Define , , then (25) holds. The proof is completed.

Remark 10. It worth noting that this manuscript consists of two important part: The modeling and algebraic formulation of the given MJNEGs; Based on the obtained algebraic expressions, we investigate the stability analysis for the given MJNEGs. Actually, this paper can be considered as a further research of [40]. Compared with the model adopted in [40], the description of MJNEGs is more general, i.e., the MJNEGs given in this paper can include the research target in [40] as a special case. In addition to that, the stability analysis for the MJNEGs in this paper is deeper than the stability analysis in [40]. We obtain better results in this paper.

4. An Illustrative Example

In this section, we use a classical example University-students game [36] to show the effective of our results.

Example 1. In an evolutionary game with random entrance, minor players have time horizon 2. denotes the maximum possible number of active players in . At the time , the number of the minor players is modelled by the vector where , is the set of players with entrance time , . Furthermore, major player plays game with minor players and minor players do not player game with each other. Thus, the random entrance is denoted by a Markov chain with states: , , and .

Define an MJNEG as follows:(i)Network topological structures, denote by , where , , , , , node 3 represents the major player, and . See in Figure 1.(ii)The FNG’s payoff is shown in Table 1.(iii)The updating rule is MBRA.(iv)Network evolve process: with a transition probability matrix

Firstly, we rewrite the MJNEG into an algebraic expression. Denote , , , network , and . Using the vector form of logical variables, we have , , , and .

With (11), one obtains

Then, by (11) and (12), we have where , , and .

Thus, by (13), one has

Solving (25), we have

Therefore, it follows from Theorem 9 that the given game is globally stable at in stochastic sense. Figure 2 demonstrates the effective of the calculation of , , and .

5. Conclusion

This paper has investigated the algebraic formulation and stability analysis for a class of MJNEG. A proper algorithm has been constructed to convert the given MJNEG into an algebraic expression. Based on the above results, the stability of the given game has been analyzed and an equivalent criterion is given. Finally, an interesting example has proved the effectiveness of our results.

In the future work, we could consider the given MJNEG with time delay. Actually, time delay is a very common situation. Many great works in the community of control and engineering have reached towards this kind of systems. See [41–60] for details.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research was supported by the Natural Science Foundation of Shandong Province under Grants ZR2018BA001 and ZR2015FQ003, the National Natural Science Foundation of China under Grants 61503225 and 61873150, and the Natural Science Fund for Distinguished Young Scholars of Shandong Province under Grant JQ201613. The material in this paper was not presented at any conference.

References

M. A. Nowak and R. M. May, “Evolutionary games and spatial chaos,” Nature, vol. 359, no. 6398, pp. 826–829, 1992.
View at: Publisher Site | Google Scholar
S. R. Gyö and A. C. Toke, “Evolutionary prisoner’s dilemma game on a square lattice,” hysical Review E, vol. 58, no. 1, pp. 69–73, 1998.
View at: Google Scholar
D. Cheng, H. Qi, and Z. Li, Analysis and Control of Boolean Network: A Semi-Tensor Product Approach, Communications and Control Engineering Series, Springer, London, UK, 2011.
View at: Publisher Site | MathSciNet
D. Z. Cheng, H. S. Qi, and Y. Zhao, “Analysis and control of Boolean networks: a matrix semi-tensor product approach,” Zidonghua Xuebao/Acta Automatica Sinica, vol. 37, no. 5, pp. 529–540, 2011.
View at: Google Scholar | MathSciNet
H. Li, Y. Wang, and L. Xie, “Output tracking control of Boolean control networks via state feedback: Constant reference signal case,” Automatica, vol. 59, article 6422, pp. 54–59, 2015.
View at: Publisher Site | Google Scholar
H. Li and Y. Wang, “Controllability analysis and control design for switched Boolean networks with state and input constraints,” SIAM Journal on Control and Optimization, vol. 53, no. 5, pp. 2955–2979, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li, L. Xie, and Y. Wang, “On robust control invariance of Boolean control networks,” Automatica, vol. 68, pp. 392–396, 2016.
View at: Publisher Site | Google Scholar
H. Li and Y. Wang, “Output tracking of switched Boolean networks under open-loop/closed-loop switching signals,” Nonlinear Analysis: Hybrid Systems, vol. 22, pp. 137–146, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li, Y. Wang, and P. Guo, “State feedback based output tracking control of probabilistic Boolean networks,” Information Sciences, vol. 349/350, pp. 1–11, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li and Y. Wang, “Minimum-time state feedback stabilization of constrained Boolean control networks,” Asian Journal of Control, vol. 18, no. 5, pp. 1688–1697, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
G. Zhao, Y. Wang, and H. Li, “Invertibility of higher order -valued logical control networks and its application in trajectory control,” Journal of The Franklin Institute, vol. 353, no. 17, pp. 4667–4679, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li and Y. Wang, “Robust stability and stabilisation of Boolean networks with disturbance inputs,” International Journal of Systems Science, vol. 48, no. 4, pp. 750–756, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li, P. Song, and Q. Yang, “Pinning control design for robust output tracking of k-valued logical networks,” Journal of The Franklin Institute, vol. 354, no. 7, pp. 3039–3053, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li, L. Xie, and Y. Wang, “Output regulation of Boolean control networks,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 62, no. 6, pp. 2993–2998, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li and Y. Wang, “Further results on feedback stabilization control design of Boolean control networks,” Automatica, vol. 83, pp. 303–308, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
X. Ding, H. Li, Q. Yang, Y. Zhou, A. Alsaedi, and F. E. Alsaadi, “Stochastic stability and stabilization of n-person random evolutionary Boolean games,” Applied Mathematics and Computation, vol. 306, pp. 1–12, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
J. Lu, H. Li, Y. Liu, and F. Li, “Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems,” IET Control Theory & Applications, vol. 11, no. 13, pp. 2040–2047, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li and Y. Wang, “Lyapunov-based stability and construction of Lyapunov functions for Boolean networks,” SIAM Journal on Control and Optimization, vol. 55, no. 6, pp. 3437–3457, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Zheng, H. Li, X. Ding, and Y. Liu, “Stabilization and set stabilization of delayed Boolean control networks based on trajectory stabilization,” Journal of The Franklin Institute, vol. 354, no. 17, pp. 7812–7827, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li, G. Zhao, M. Meng, and J. Feng, “A survey on applications of semi-tensor product method in engineering,” Science China Information Sciences, vol. 61, no. 1, 010202, 17 pages, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Li, W. Dou, H. Li, and X. Liu, “Controllability, Reachability, and Stabilizability of Finite Automata: A Controllability Matrix Method,” Mathematical Problems in Engineering, vol. 2018, Article ID 6719319, 6 pages, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
S. Fu, H. Li, and G. Zhao, “Modelling and strategy optimisation for a kind of networked evolutionary games with memories under the bankruptcy mechanism,” International Journal of Control, vol. 91, no. 5, pp. 1104–1117, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Li, H. Li, and P. Duan, “Synchronization of switched logical control networks via event-triggered control,” Journal of The Franklin Institute, vol. 355, no. 12, pp. 5203–5216, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
S. Fu, G. Zhao, and H. Li, “Model and control for a class of networked evolutionary games with finite memories and time-varying networks,” Circuits Systems & Signal Processing, vol. 37, no. 7, pp. 3093–3114, 2018.
View at: Publisher Site | Google Scholar
Y. Li, H. Li, X. Ding, and G. Zhao, “Leader-follower consensus of multi-agent systems with time delays over finite fields,” IEEE Transactions on Cybernetics, 2018.
View at: Google Scholar
Y. Li, H. Li, and W. Sun, “Event-triggered control for robust set stabilization of logical control networks,” Automatica, vol. 95, pp. 556–560, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Li, H. Li, and X. Xu, “A semi-tensor product approach to minimal-agent consensus control of networked evolutionary games,” IET Control Theory Applications, pp. 10–1049, 2018.
View at: Google Scholar
G. Zhao and S. Fu, “Matrix approach to trajectory control of higher-order -valued logical control networks,” IET Control Theory & Applications, vol. 11, no. 13, pp. 2110–2115, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
G. Zhao, H. Li, W. Sun, and F. . Alsaadi, “Modelling and strategy consensus for a class of networked evolutionary games,” International Journal of Systems Science, vol. 49, no. 12, pp. 2548–2557, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
P. Guo, Y. Wang, and H. Li, “Algebraic formulation and strategy optimization for a class of evolutionary networked games via semi-tensor product method,” Automatica, vol. 49, no. 11, pp. 3384–3389, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
D. Cheng, F. He, H. Qi, and T. Xu, “Modeling, analysis and control of networked evolutionary games,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 60, no. 9, pp. 2402–2415, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
G. Zhao and Y. Wang, “Formulation and optimization control of a class of networked evolutionary games with switched topologies,” Nonlinear Analysis: Hybrid Systems, vol. 22, pp. 98–107, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
B. Zhu, X. Xia, and Z. Wu, “Evolutionary game theoretic demand-side management and control for a class of networked smart grid,” Automatica, vol. 70, pp. 94–100, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
P. Guo, H. Zhang, F. E. Alsaadi, and T. Hayat, “Semi-tensor product method to a class of event-triggered control for finite evolutionary networked games,” IET Control Theory & Applications, vol. 11, no. 13, pp. 2140–2145, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
I. Kordonis and G. P. Papavassilopoulos, “LQ Nash games with random entrance: an infinite horizon major player and minor players of finite horizons,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 60, no. 6, pp. 1486–1500, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
G. P. Papavassilopoulos, “University-students game,” Dynamic Games and Applications, vol. 3, no. 3, pp. 387–418, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
H. Li, Y. Wang, and Z. Liu, “A semi-tensor product approach to pseudo-BOOlean functions with application to BOOlean control networks,” Asian Journal of Control, vol. 16, no. 4, pp. 1073–1081, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
G. Ellison, “Learning, local interaction, and coordination,” Econometrica, vol. 61, no. 5, pp. 1047–1071, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
X. Chen, J. Lam, P. Li, and Z. Shu, “-induced norm and controller synthesis of positive systems,” Automatica, vol. 49, no. 5, pp. 1377–1385, 2013.
View at: Publisher Site | Google Scholar
G. Zhao, Y. Wang, and H. Li, “A matrix approach to modeling and optimization for dynamic games with random entrance,” Applied Mathematics and Computation, vol. 290, pp. 9–20, 2016.
View at: Publisher Site | Google Scholar
X. Li and J. Wu, “Stability of nonlinear differential systems with state-dependent delayed impulses,” Automatica, vol. 64, pp. 63–69, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
X. Li and S. Song, “Stabilization of Delay Systems: delay-dependent Impulsive Control,” IEEE Transactions on Automatic Control, vol. 62, no. 1, pp. 406–411, 2017.
View at: Publisher Site | Google Scholar
X. Zhang and X. Li, “Input-to-state stability of non-linear systems with distributed-delayed impulses,” IET Control Theory & Applications, vol. 11, no. 1, pp. 81–89, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
R. Rakkiyappan, G. Velmurugan, X. Li, and D. O’Regan, “Global dissipativity of memristor-based complex-valued neural networks with time-varying delays,” Neural Computing and Applications, vol. 27, no. 3, pp. 629–649, 2016.
View at: Publisher Site | Google Scholar
X. Li, X. Zhang, and S. Song, “Effect of delayed impulses on input-to-state stability of nonlinear systems,” Automatica, vol. 76, pp. 378–382, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
X. Li and J. Cao, “An impulsive delay inequality involving unbounded time-varying delay and applications,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 62, no. 7, pp. 3618–3625, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
X. Li and J. Wu, “Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 63, no. 1, pp. 306–311, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
X. Li and Y. Ding, “Razumikhin-type theorems for time-delay systems with Persistent impulses,” Systems & Control Letters, vol. 107, pp. 22–27, 2017.
View at: Publisher Site | Google Scholar
J. Hu, G. Sui, S. Du, and X. Li, “Finite-Time Stability of Uncertain Nonlinear Systems with Time-Varying Delay,” Mathematical Problems in Engineering, vol. 2017, Article ID 2538904, 9 pages, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
X. Zhang, X. Lv, and X. Li, “Sampled-data-based lag synchronization of chaotic delayed neural networks with impulsive control,” Nonlinear Dynamics, vol. 90, no. 3, pp. 2199–2207, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
X. Lv and X. Li, “Finite time stability and controller design for nonlinear impulsive sampled-data systems with applications,” ISA Transactions, vol. 70, pp. 30–36, 2017.
View at: Publisher Site | Google Scholar
E. A. Alzahrani, H. Akca, and X. Li, “New synchronization schemes for delayed chaotic neural networks with impulses,” Neural Computing and Applications, vol. 28, no. 9, pp. 2823–2837, 2017.
View at: Publisher Site | Google Scholar
H. Zhang, R. Ye, J. Cao, A. Ahmed, X. Li, and Y. Wan, “Lyapunov Functional Approach to Stability Analysis of Riemann-Liouville Fractional Neural Networks with Time-Varying Delays,” Asian Journal of Control, vol. 20, no. 5, pp. 1938–1951, 2018.
View at: Publisher Site | Google Scholar
X. Li, P. Li, and Q. Wang, “Input/output-to-state stability of impulsive switched systems,” Systems Control Letters, p. 116, 2018.
View at: Google Scholar
X. Yang, X. Li, and J. Cao, “Robust finite-time stability of singular nonlinear systems with interval time-varying delay,” Journal of The Franklin Institute, vol. 355, no. 3, pp. 1241–1258, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
X. Tan, J. Cao, X. Li, and A. Alsaedi, “Leader-following mean square consensus of stochastic multi-agent systems with input delay via event-triggered control,” IET Control Theory & Applications, vol. 12, no. 2, pp. 299–309, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
X. Tan, J. Cao, and X. Li, “Consensus of Leader-Following Multiagent Systems: A Distributed Event-Triggered Impulsive Control Strategy,” IEEE Transactions on Cybernetics, 2018.
View at: Google Scholar
X. Lv, R. Rakkiyappan, and X. Li, “μ-stability criteria for nonlinear differential systems with additive leakage and transmission time-varying delays,” Nonlinear Analysis: Modelling and Control, vol. 23, no. 3, pp. 380–400, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Wang, J. Cao, X. Li, and A. Alsaedi, “Edge-based epidemic dynamics with multiple routes of transmission on random networks,” Nonlinear Dynamics, vol. 91, no. 1, pp. 403–420, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
B. S. Vadivoo, R. Ramachandran, J. Cao, H. Zhang, and X. Li, “Controllability analysis of nonlinear neutral-type fractional-order differential systems with state delay and impulsive effects,” International Journal of Control, Automation and Systems, pp. 1–11, 2018.
View at: Google Scholar

Copyright

Copyright © 2018 Guodong Zhao. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

517

Downloads

604

Citations