# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

April 2021 , Volume 14 , Issue 4

Issue on dynamics and control of complex systems

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2021, 14(4): i-ii doi: 10.3934/dcdss.2021024 +[Abstract](211) +[HTML](90) +[PDF](69.27KB)
Abstract:
2021, 14(4): 1213-1231 doi: 10.3934/dcdss.2020376 +[Abstract](806) +[HTML](328) +[PDF](579.78KB)
Abstract:

An approach to output feedback control for hybrid discrete-time systems subject to uncertain mode transitions is proposed. The system dynamics may assume different modes upon the occurrence of a switching that is not directly measurable. Since the current system mode is unknown, a regulation scheme is proposed by combining a Luenberger observer to estimate the continuous state, a mode estimator, and a controller fed with the estimates of both continuous state variables and mode. The closed-loop stability is ensured under suitable conditions given in terms of linear matrix inequalities. Since complexity and conservativeness grow with the increase of the modes, we address the problem of reducing the number of linear matrix inequalities by providing more easily tractable stability conditions. Such conditions are extended to deal with systems having also Lipschitz nonlinearities and affected by disturbances. The effectiveness of the proposed approach is shown by means of simulations.

2021, 14(4): 1233-1244 doi: 10.3934/dcdss.2020356 +[Abstract](1181) +[HTML](381) +[PDF](508.93KB)
Abstract:

In this paper we propose a stochastic mathematical model with distributed delay in order to describe the transmission dynamics of cocaine consumption in Spain. We investigate conditions to guarantee the stability in probability of the equilibrium points under stochastic perturbations via the white noise processes. The results are applied to the model cocaine consumption using data retrieved from the Spanish Drug National Plan, http://www.pnsd.mscbs.gob.es/. The obtained results may be useful for policy health authorities in order to improve the strategies against the drug consumption in the long-run.

2021, 14(4): 1245-1258 doi: 10.3934/dcdss.2020359 +[Abstract](864) +[HTML](361) +[PDF](283.43KB)
Abstract:

The key contribution of this paper is to study the stability analysis of neutral-type Cohen-Grossberg neural networks possessing multiple time delays in the states of the neurons and multiple neutral delays in time derivative of states of the neurons. By making the use of a proper Lyapunov functional, we propose a novel sufficient time-independent stability criterion for this model of neutral-type neural networks. The proposed stability criterion in this paper can be absolutely expressed in terms of the parameters of the neural network model considered as this newly proposed criterion only relies on the relationships established among the network parameters. A numerical example is also given to indicate the advantages of the obtained stability criterion over the previously published stability results for the same class of Cohen-Grossberg neural networks. Since obtaining stability conditions for neutral-type Cohen-Grossberg neural networks with multiple delays is a difficult task to achieve, there are only few papers in the literature dealing with this problem. Therefore, the results given in the current paper makes an important contribution to the stability problem for this class of neutral-type neural networks.

2021, 14(4): 1259-1272 doi: 10.3934/dcdss.2020372 +[Abstract](898) +[HTML](331) +[PDF](444.2KB)
Abstract:

Taking into account the effects of multi-proportional delays and D operator, this paper investigates the stability issue of a general class of neutral-type SICNNs (shunting inhibitory cellular neural networks). With the help of fixed point theorem and some novel differential inequality techniques, we derive a new sufficient conditions to ensure the existence, uniqueness and exponential stability of weighted pseudo almost periodic solutions (WPAPS) of the considered model. The obtained main results are totally new and generalize some published results. At the end of this work, we also give some numerical simulations to support the proposed approach and demonstrate the correctness of the main conclusions.

2021, 14(4): 1273-1299 doi: 10.3934/dcdss.2020369 +[Abstract](775) +[HTML](358) +[PDF](784.19KB)
Abstract:

In the past decades, complex systems with impulsive effects and logical dynamics have received much attention in both the natural and social sciences. This historical survey briefly introduces relevant studies on impulsive differential systems (IDSs) and logical networks (LNs), respectively. To begin with, we investigate five aspects of IDSs containing fundamental theory, Lyapunov stability, input-to-state stability, hybrid impulses and delay-dependent impulses. Next, we compactly summarize the research status of some problems of LNs including controllability, stability and stabilization, observability and current research. Moreover, some significant applications of proposed results are illustrated. Finally, based on this overview, we further discuss some future work on complex systems with impulsive effects and logical dynamics.

2021, 14(4): 1301-1328 doi: 10.3934/dcdss.2020412 +[Abstract](655) +[HTML](321) +[PDF](481.78KB)
Abstract:

This paper investigates the event-based fault detection (FD) problem for a category of discrete-time interval type-2 fuzzy systems with measurement outliers. For the sake of decreasing the utilization of limited communication bandwidth, an event-based mechanism is introduced. Based on the saturation function technique, a novel event-based FD observer is first designed to reduce the influence of outliers in the dynamic systems. Then, on the basis of Lyapunov stability theory, sufficient conditions are provided to ensure that the error system satisfies the \begin{document}$\mathcal{H}_{\infty}$\end{document} performance and the \begin{document}$\mathcal{H}_{\infty}$\end{document} fault performance in different cases, respectively. In contrast to the existing event-based FD results, the false alarm, which is induced by measurement outliers, can be effectively avoided by the designed FD observer with saturation function. Lastly, some simulation results are given to verify the effectiveness of the method presented in this paper.

2021, 14(4): 1329-1343 doi: 10.3934/dcdss.2020368 +[Abstract](746) +[HTML](296) +[PDF](405.57KB)
Abstract:

This paper investigates the problems of \begin{document}$\mathcal{H}_{\infty}$\end{document} performance analysis and sampled-data control about fuzzy Markovian jump systems. Firstly, in order to make full use of the information of both intervals \begin{document}$x(t_{k})$\end{document} to \begin{document}$x(t)$\end{document} and \begin{document}$x(t)$\end{document} to \begin{document}$x(t_{k+1})$\end{document}, we construct the mode-dependent Lyapunov function, which consists of a two-sided closed-loop function. Built on the above Lyapunov function, the stochastically stable conditions with less conservative are given by using linear matrices inequalities (LMIs). Then, a state feedback controller is presented for the studied systems. At last, an example is offered to illustrate the efficiency of our main results.

2021, 14(4): 1345-1358 doi: 10.3934/dcdss.2020367 +[Abstract](688) +[HTML](282) +[PDF](455.61KB)
Abstract:

We show the existence of an exponential attractor for non-autono-mous dynamical system with bounded delay. We considered the case of strong dissipativity then prove that the result remains for the weak dissipativity. We conclude then the existence of the global attractor and ensure the boundedness of its fractal dimension.

2021, 14(4): 1359-1373 doi: 10.3934/dcdss.2020371 +[Abstract](2133) +[HTML](318) +[PDF](5973.34KB)
Abstract:

Recently, quantifying the level of the synchrony in non-identical networks has got considerable attention. In the first part of this paper, a new synchronization index for non-identical networks is proposed. Non-identical networks can be categorized into two main types. The first group consists of similar oscillators with miss-match in their parameters, and the second group is organized from completely different oscillators. The synchronizability of the second group of the non-identical networks is more challenging since the amplitude and frequencies of the different oscillators may not be matched. Thus, one way to investigate the limitation of the synchronizability of these networks is to explore the parameter space of their amplitude and frequency. In the second part of this research, the amplitude and frequency of each individual system of the non-identical network are considered as varying parameters and the effect of these parameters on the synchronizability of the network is measured with the propsed index. The results are compared with the conventional indexes, such as the root-mean-square error and phase synchrony with the help of Hilbert transform. The outcomes show that the new proposed synchronization index not only is simple and accurate, but also fast with short computational time. It is not affected by amplitude, phase, or polarity. It can detect the similarity in the fluctuations which is a sign of synchrony in the non-identical networks.

2021, 14(4): 1375-1393 doi: 10.3934/dcdss.2020280 +[Abstract](1112) +[HTML](450) +[PDF](416.55KB)
Abstract:

The stochastic inflow or withdrawal of funds in the international financial market are both positive or negative external inputs to the domestic financial system. So, in this paper, input-to-state stability criterion of delayed feedback chaotic financial system is investigated, and derived by counterevidence method, Lyapunov functional method, variational method and regional control technique, which was involved to equilibrium solution with the positive interest rate. On the other hand, if these inputs are too small to be ignored, impulse control can be applied to stability analysis of the delayed feedback system, in which the delayed impulse allows the pulse effect to lag for a period of time. The obtained stability criteria show that no matter how complex and chaos the financial system is, high-frequency effective macro-control is conducive to the global asymptotical stability of the economic system, including the open economic model with foreign investment fund inputs. Finally, numerical examples illustrate the effectiveness of all the proposed methods.

2021, 14(4): 1395-1414 doi: 10.3934/dcdss.2020379 +[Abstract](801) +[HTML](403) +[PDF](2143.4KB)
Abstract:

This paper presents an event-triggered consensus control protocol for a class of multi-agent systems with actuator faults, sensor faults and unknown disturbances. The adaptive neural network compensation control method is introduced to solve the problem of sensor faults. The event-triggered mechanism is developed to reduce the communication burden. In the control design process, the radial basis function neural networks are used to approximate the unknown nonlinear functions, and a nonlinear disturbance observer is used to eliminate the effect of unknown external disturbances. Furthermore, based on the graph theory and Lyapunov stability theory, it is further shown that the consensus tracking errors are semi-globally uniformly ultimately bounded. Finally, the simulation example illustrates the effectiveness of the designed control protocol.

2021, 14(4): 1415-1428 doi: 10.3934/dcdss.2020358 +[Abstract](823) +[HTML](330) +[PDF](274.52KB)
Abstract:

It is well-known that the global asymptotic stability analysis of neutral systems is an important concept in designing the appropriate controllers or filters for this class of systems. This paper carries out a delay-independent stability analysis of neutral systems possessing discrete time delays in the states and discrete neutral delays in the time derivative of the states in the presence of nonlinear disturbances. Some new global asymptotic stability criteria are proposed by introducing a novel Lyapunov functional. The obtained delay-independent stability criteria establish some simple and easily verifiable mathematical expressions involving the elements of the system matrices and the disturbance parameters of the neutral system. Different from the most of the previously reported stability results for neutral systems, the conditions obtained in this paper are not expressed in terms the Linear Matrix Inequalities (LMIs). Therefore, the criteria presented in this paper can be considered as the alternative results to previously published stability results stated in the LMI forms. A comparison between the results of this paper and some of previously published corresponding stability results is made to substantiate the significant improvement of the proposed results. A constructive numerical example is also presented to show applicability and the effectiveness of the proposed stability condition.

2021, 14(4): 1429-1446 doi: 10.3934/dcdss.2020370 +[Abstract](656) +[HTML](294) +[PDF](360.41KB)
Abstract:

In this paper, we introduce the notion of stability of sets for reaction-diffusion Cohen–Grossberg neural networks with time-varying delays. The Lyapunov–Razumikhin technique and a comparison principle are adapted to prove the new stability criteria. In addition, the obtained results are extended to the uncertain case, and the robust stability notion is also investigated. Examples are considered to demonstrate the effectiveness of our results.

2021, 14(4): 1447-1464 doi: 10.3934/dcdss.2020375 +[Abstract](761) +[HTML](288) +[PDF](1490.99KB)
Abstract:

In an electronically controlled VE distributive pump, the fuel quantity actuator is a significant component. It is responsible for governing the quantity of fuel being injected into diesel-type engines. The FQA system has nonlinearities and always confronts disturbances caused by the external torque and the input voltage variation in the real working condition, which can be regarded as a lumped disturbance. However, most existing results only focus on dealing with the so called constant disturbance in the FQA system which fail to remove the influence of time-varying disturbances. Therefore, to deal with the nonlinearities and reject the lumped disturbance, a reduced-order generalized proportional integral observer (GPIO) based sliding mode control approach is presented. By using a reduced-order GPIO, time-varying disturbances can be estimated accurately. In addition, a theoretical analysis of the closed-loop system is given. The proposed control scheme exhibits a satisfactory performance in terms of transient behavior and disturbance rejection. Finally, a set of experimental tests are carried out to validate the feasibility as well as efficiency of the proposed control framework.

2021, 14(4): 1465-1477 doi: 10.3934/dcdss.2020395 +[Abstract](679) +[HTML](323) +[PDF](399.55KB)
Abstract:

This investigation looks at the issue of finite time exponential synchronization of complex dynamical systems with reaaction diffusion term. This reort studies complex networks consisting of \begin{document}$N$\end{document} straightly and diffusively coupled networks. By building a new Lyapunov krasovskii functional (LKF), using Jensens inequality and convex algorithms approach stability conditions frameworks are determined. At last, a numerical precedent is given to demonstrate the practicality of the theoretical results.

2021, 14(4): 1479-1494 doi: 10.3934/dcdss.2020374 +[Abstract](941) +[HTML](300) +[PDF](2249.89KB)
Abstract:

This paper studies synchronization phenomena of spike-trains and approximation of target spike-trains in a simple network of digital spiking neurons. Repeating integrate-and-fire behavior between a periodic base signal and constant firing threshold, the neurons can generate various spike-trains. Connecting multiple neurons by cross-firing with delay, the network is constructed. The network can exhibit multi-phase synchronization of various spike-trains. Stability of the synchronization phenomena can be guaranteed theoretically. Applying a simple winner-take-all switching, the network can approximate target spike-trains automatically. In order to evaluate the approximation performance, we present two metrics: spike-position error and spike missing rate. Using the metrics, approximation capability of the network is investigated in typical target signals. Presenting an FPGA based hardware prototype, typical synchronization phenomenon and spike-train approximation are confirmed experimentally.

2021, 14(4): 1495-1518 doi: 10.3934/dcdss.2020377 +[Abstract](770) +[HTML](325) +[PDF](2062.19KB)
Abstract:

In this paper, an event-triggered reinforcement learning-based met-hod is developed for model-based optimal synchronization control of multiple Euler-Lagrange systems (MELSs) under a directed graph. The strategy of event-triggered optimal control is deduced through the establishment of Hamilton-Jacobi-Bellman (HJB) equation and the triggering condition is then proposed. Event-triggered policy iteration (PI) algorithm is then borrowed from reinforcement learning algorithms to find the optimal solution. One neural network is used to represent the value function to find the analytical solution of the event-triggered HJB equation, weights of which are updated aperiodically. It is proved that both the synchronization error and the weight estimation error are uniformly ultimately bounded (UUB). The Zeno behavior is also excluded in this research. Finally, an example of multiple 2-DOF prototype manipulators is shown to validate the effectiveness of our method.

2021, 14(4): 1519-1533 doi: 10.3934/dcdss.2020373 +[Abstract](769) +[HTML](319) +[PDF](516.89KB)
Abstract:

This paper addresses the observability of a switched Boolean control network (SBCN) using semi-tensor product (STP) of matrices. First, the observability of the SBCN is determined under desirable switching signals and arbitrary switching signals by encoding the switching signal as a boolean input. Then an algorithm is designed for determining the observability. Furthermore, feedback control laws are obtained to guarantee the observability of SBCNs. Examples and corresponding state trajectory graphs are given to illustrate the effectiveness of the given results.

2021, 14(4): 1535-1551 doi: 10.3934/dcdss.2020396 +[Abstract](925) +[HTML](290) +[PDF](377.61KB)
Abstract:

In this paper, the distributed consensus problem is investigated for a class of higher-order nonlinear multi-agent systems with unmatched disturbances. By the back-stepping technique, a new distributed protocol is designed to solve the consensus problem for multi-agent systems without using the information of the Laplacian matrix and Lipschitz constants. It is proved that the practical consensus of multi-agent systems with unmatched disturbances can be achieved by the proposed protocol. Finally, the validity of the proposed scheme is verified by a simulation.

2021, 14(4): 1553-1567 doi: 10.3934/dcdss.2020378 +[Abstract](920) +[HTML](306) +[PDF](406.33KB)
Abstract:

This paper investigates the optimization of traffic congestion systems via network congestion game approach. Firstly, using the semi-tensor product(STP) of matrices, the matrix expression of network congestion game is obtained. Secondly, a necessary and sufficient condition is proposed to guarantee that the traffic systems can be transformed into network congestion game with given performance criterion as its weighted potential function. Then an algorithm is provided to design the traffic congestion price in the case that conversion can be established. Thirdly, by designing proper learning rule, the optimization of traffic systems can be achieved when individuals optimize their own utility function. Moreover, two special cases which make our results more accord with reality and rich. Finally, an example is exploited to demonstrate the effectiveness of our obtained results.

2021, 14(4): 1569-1589 doi: 10.3934/dcdss.2020357 +[Abstract](944) +[HTML](346) +[PDF](774.83KB)
Abstract:

This paper is concerned with the issue of fault-tolerant anti-synchro-nization control for chaotic switched neural networks with time delay and reaction-diffusion terms under the drive-response scheme, where the response system is assumed to be disturbed by stochastic noise. Both arbitrary switching signal and average dwell-time limited switching signal are taken into account. With the aid of the Lyapunov-Krasovskii functional approach and combining with the generalized Itô formula, sufficient conditions on the mean-square exponential stability for the anti-synchronization error system are presented. Then, by utilizing some decoupling methods, constructive design strategies on the desired fault-tolerant anti-synchronization controller are proposed. Finally, an example is given to demonstrate the effectiveness of our design strategies.

2021, 14(4): 1591-1605 doi: 10.3934/dcdss.2020413 +[Abstract](666) +[HTML](288) +[PDF](392.92KB)
Abstract:

In this paper, the set stabilization problem of Markovian jump Boolean control networks (MJBCNs) is investigated via semi-tensor product of matrices. First, the conception of set stabilization is proposed for MJBCNs. Then based on the algebraic expression of MJBCN, a necessary and sufficient condition for set stabilization is provided by a linear programming problem, which is simple to solve. Moreover, by solving this linear programming problem, an algorithm for designing a state feedback controller is developed. Finally, two examples are presented to illustrate the feasibility of the obtained results.

2021, 14(4): 1607-1629 doi: 10.3934/dcdss.2020279 +[Abstract](980) +[HTML](481) +[PDF](532.67KB)
Abstract:

This paper concerns the synchronization of a kind of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations. Multi-layer networks are a kind of complex networks with different layers, which consist of different kinds of interactions or multiple subnetworks. Additive couplings are designed to capture the different layered connections. In this paper, two pinning controllers are designed to guarantee the synchronization of the stochastic multi-layer network. One is the state-feedback pinning controller with constant control gains. The other one is the adaptive pinning controller with adaptive control gains. It is worthwhile to mention that our assumptions on the activation functions satisfy a generalized Lipschitzian condition which are weaker than those in the previous works. Moreover, as we prove, only selected part of the nodes to be controlled are enough to guarantee that the drive system and response network can be stochastically synchronized. Finally, an example and its simulations are presented to show the feasibility effectiveness of our control schemes.

2019  Impact Factor: 1.233