Editorial

4 pages, 171 KiB

Open AccessEditorial

Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value

by Xinsong Yang and Ruofeng Rao

Mathematics 2023, 11(10), 2247; https://doi.org/10.3390/math11102247 - 11 May 2023

Viewed by 671

Well-posedness, dynamics, and control of nonlinear differential system with an initial-boundary value involve many mathematical, physical, and engineering problems [...] Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

Research

Jump to: Editorial

16 pages, 5619 KiB

Open AccessArticle

Adaptive Output Feedback Control for Constrained Switched Systems with Input Quantization

by Shuyan Qi, Jun Zhao and Li Tang

Mathematics 2023, 11(3), 788; https://doi.org/10.3390/math11030788 - 03 Feb 2023

Cited by 4 | Viewed by 1047

Abstract

This paper investigates adaptive output feedback control problem for switched uncertain nonlinear systems with input quantization, unmeasured system states and state constraints. Firstly, fuzzy logic systems are introduced to identify system uncertainties, then the fuzzy based observer is constructed to estimate unavailable states. [...] Read more.

This paper investigates adaptive output feedback control problem for switched uncertain nonlinear systems with input quantization, unmeasured system states and state constraints. Firstly, fuzzy logic systems are introduced to identify system uncertainties, then the fuzzy based observer is constructed to estimate unavailable states. Secondly, combing the backstepping technique and the barrier Lyapunov function method, an adaptive fuzzy output feedback control law is designed, which guarantees that all signals in the closed-loop system are bounded, the system output tracks the reference signal, and system states satisfy their corresponding constraint conditions. Finally, simulation results further show the good performances of the proposed control scheme. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

14 pages, 2023 KiB

Open AccessArticle

Complex Dynamical Sampling Mechanism for the Random Pulse Circulation Model and Its Application

by Lin Tang, Kaibo Shi and Songke Yu

Mathematics 2023, 11(3), 668; https://doi.org/10.3390/math11030668 - 28 Jan 2023

Viewed by 784

Abstract

The fast multi-pulse spectrum is a spectrum acquisition method that obtains an average pulse amplitude in a dynamic window, which improves the energy resolution by sharpening peaks in the acquired spectra, but produces the counting loss. Owing to the counting loss problem, a [...] Read more.

The fast multi-pulse spectrum is a spectrum acquisition method that obtains an average pulse amplitude in a dynamic window, which improves the energy resolution by sharpening peaks in the acquired spectra, but produces the counting loss. Owing to the counting loss problem, a counting rate multiplication method based on uniform sampling, also called the pulse circulation method, is presented in this paper. Based on the theory of mathematical statistics and uniform sampling, this method adopted a dynamic sample pool to update the pulse amplitude sample in real time. Random numbers from the uniform distribution were sampled from the sample pool, and the sampled results were stored in the random pulse circulator so that the pulse amplitude information used for spectrum generation was uniformly expanded. In the experiment section, the obtained spectrum was analyzed to verify the multiplication effect of the pulse circulation method on the counting rate and the compensation effect of the fast multi-pulse spectrum algorithm on the counting rate loss. The results indicated that the characteristic peaks of each element in the X-ray spectrogram obtained by the pulse circulation method could realize counting rate multiplication uniformly, and the multiplication ratio of every element was approximately equal. This is of great significance for obtaining an accurate X-ray fluorescence spectrum. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

12 pages, 1008 KiB

Open AccessArticle

Impulsive Control of Some Types of Nonlinear Systems Using a Set of Uncertain Control Matrices

by Keke Wu, Babatunde Oluwaseun Onasanya, Longzhou Cao and Yuming Feng

Mathematics 2023, 11(2), 421; https://doi.org/10.3390/math11020421 - 13 Jan 2023

Cited by 3 | Viewed by 977

Abstract

So many real life problems ranging from medicine, agriculture, biology and finance are modelled by nonlinear systems. In this case, a chaotic nonlinear system is considered and, as opposed to solving Linear Matrix Inequality (LMI), which is the usual approach but cumbersome, a [...] Read more.

So many real life problems ranging from medicine, agriculture, biology and finance are modelled by nonlinear systems. In this case, a chaotic nonlinear system is considered and, as opposed to solving Linear Matrix Inequality (LMI), which is the usual approach but cumbersome, a completely different approach was used. In some other cases, the computation of singular value of matrix was used but the method in this study needs not such. In addition, most models, if not all, concentrate on finding a control matrix J under some sufficient conditions. The problem is that only one such matrix J is provided. In reality, the actual control quantity may have a little deviation from the theoretical J. Hence, the study in this paper provides a set of infinite uncertain matrices

J_{α}

which are able to adapt to control the system under uncertain conditions. It turns out that this new method controls the system in shorter time with less computational complexities. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

23 pages, 3224 KiB

Open AccessArticle

Stability and Synchronization of Fractional-Order Complex-Valued Inertial Neural Networks: A Direct Approach

by Hualin Song, Cheng Hu and Juan Yu

Mathematics 2022, 10(24), 4823; https://doi.org/10.3390/math10244823 - 19 Dec 2022

Cited by 4 | Viewed by 1200

Abstract

This paper is dedicated to the asymptotic stability and synchronization for a type of fractional complex-valued inertial neural network by developing a direct analysis method. First, a new fractional differential inequality is presented for nonnegative functions, which provides an effective tool for the [...] Read more.

This paper is dedicated to the asymptotic stability and synchronization for a type of fractional complex-valued inertial neural network by developing a direct analysis method. First, a new fractional differential inequality is presented for nonnegative functions, which provides an effective tool for the convergence analysis of fractional-order systems. Moreover, instead of the previous separation analysis for complex-valued neural networks, a class of Lyapunov functions composed of the complex-valued states and their fractional derivatives is constructed, and some compact stability criteria are derived. In synchronization analysis, unlike the existing control schemes for reduced-order subsystems, some feedback and adaptive control schemes, formed by the linear part and the fractional derivative part, are directly designed for the response fractional inertial neural networks, and some synchronization conditions are derived using the established fractional inequality. Finally, the theoretical analysis is supported via two numerical examples. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

14 pages, 487 KiB

Open AccessArticle

An Uncertain Sandwich Impulsive Control System with Impulsive Time Windows

by Huafei Chen, Jia Chen, Dan Qu, Kelin Li and Fei Luo

Mathematics 2022, 10(24), 4708; https://doi.org/10.3390/math10244708 - 12 Dec 2022

Cited by 4 | Viewed by 1030

Abstract

In this paper, we formulate a new system, named the uncertain sandwich impulsive control system with impulsive time windows. The presented system shows that the linear entry matrix of the system is indeterminate. We first investigate the exponential stability of the considered system [...] Read more.

In this paper, we formulate a new system, named the uncertain sandwich impulsive control system with impulsive time windows. The presented system shows that the linear entry matrix of the system is indeterminate. We first investigate the exponential stability of the considered system by linear matrix inequalities (LMIs) and inequalities techniques, then extend the considered system to a more general one and further study the stability of the general system. Finally, numerical simulations are delivered to demonstrate the effectiveness of the theoretical results. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

16 pages, 1752 KiB

Open AccessArticle

Three-Stage-Impulse Control of Memristor-Based Chen Hyper-Chaotic System

by Xianyang Xie, Shiping Wen, Yuming Feng and Babatunde Oluwaseun Onasanya

Mathematics 2022, 10(23), 4560; https://doi.org/10.3390/math10234560 - 01 Dec 2022

Cited by 7 | Viewed by 1109

Abstract

In this paper, on the basis of the three-dimensional Chen system, a smooth continuous nonlinear flux-controlled memristor model is used as the positive feedback term of this system, a hyper-chaotic circuit system is successfully constructed, and a simulated equivalent circuit is built for [...] Read more.

In this paper, on the basis of the three-dimensional Chen system, a smooth continuous nonlinear flux-controlled memristor model is used as the positive feedback term of this system, a hyper-chaotic circuit system is successfully constructed, and a simulated equivalent circuit is built for simulation using Multisim software, which agrees with the numerical simulation results by comparison. Meanwhile, a new impulsive control mode called the three-stage-impulse is put forward. It is a cyclic system with three components: continuous inputs are exerted in the first and third parts of the cycle while giving no input in the second part of the cycle, an impulse is exerted at the end of each continuous subsystem, the controller is simple in structure and effective in stabilizing most existing nonlinear systems. The Chen hyper-chaotic system will be controlled based on the three-stage-impulse control method combined with the Lyapunov stability principle. At the end of this paper, we have employed and simulated a numerical example; the experimental results show that the controller is effective for controlling and stabilizing the newly designed hyper-chaotic system. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

11 pages, 291 KiB

Open AccessArticle

A New Slack Lyapunov Functional for Dynamical System with Time Delay

by Can Zhao, Kaibo Shi, Yiqian Tang and Shouming Zhong

Mathematics 2022, 10(23), 4462; https://doi.org/10.3390/math10234462 - 26 Nov 2022

Cited by 1 | Viewed by 834

Abstract

The traditional method of constructing a Lyapunov functional for dynamical systems with time delay is usually dependent on positive definite matrices in the quadratic form. In this paper, a new Lyapunov functional is proposed and the corresponding proof is given. It do not [...] Read more.

The traditional method of constructing a Lyapunov functional for dynamical systems with time delay is usually dependent on positive definite matrices in the quadratic form. In this paper, a new Lyapunov functional is proposed and the corresponding proof is given. It do not require that all matrices in the quadratic form of Lyapunov functionals are positive definite, while the quadratic form is still positive definite, which makes the estimate more relaxed due to special construction of matrices. It is a general form and can be used in the performance analysis of a variety of dynamical systems. Moreover, a lemma concerning the quadratic function is applied to deal with the quadratic term of time-varying delay. Lastly, in the case of classical dynamical systems with time delay, the verification results are given to illustrate the usefulness of the new slack Lyapunov functional. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

15 pages, 540 KiB

Open AccessArticle

Boundary Coupling for Consensus of Nonlinear Leaderless Stochastic Multi-Agent Systems Based on PDE-ODEs

by Chuanhai Yang, Jin Wang, Shengfa Miao, Bin Zhao, Muwei Jian and Chengdong Yang

Mathematics 2022, 10(21), 4111; https://doi.org/10.3390/math10214111 - 04 Nov 2022

Cited by 1 | Viewed by 1035

Abstract

This paper studies the leaderless consensus of the stochastic multi-agent systems based on partial differential equations–ordinary differential equations (PDE-ODEs). Compared with the traditional state coupling, the most significant difference between this paper is that the space state coupling is designed. Two boundary couplings [...] Read more.

This paper studies the leaderless consensus of the stochastic multi-agent systems based on partial differential equations–ordinary differential equations (PDE-ODEs). Compared with the traditional state coupling, the most significant difference between this paper is that the space state coupling is designed. Two boundary couplings are investigated in this article, respectively, collocated boundary measurement and distributed boundary measurement. Using the Lyapunov directed method, sufficient conditions for the stochastic multi-agent system to achieve consensus can be obtained. Finally, two simulation examples show the feasibility of the proposed spatial boundary couplings. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

24 pages, 1188 KiB

Open AccessFeature PaperArticle

Improved Nonfragile Sampled-Data Event-Triggered Control for the Exponential Synchronization of Delayed Complex Dynamical Networks

by Can Zhao, Jinde Cao, Kaibo Shi, Yiqian Tang, Shouming Zhong and Fawaz E. Alsaadi

Mathematics 2022, 10(19), 3504; https://doi.org/10.3390/math10193504 - 26 Sep 2022

Cited by 3 | Viewed by 1214

Abstract

The exponential synchronization of complex dynamical networks (CDNs) under improved nonfragile sampled-data event-triggered control (INFSDETC) is investigated in this study. A meaningful yet challenging issue is solved, namely, it can adjust the triggering mode and the triggering frequency to adapt to more situations [...] Read more.

The exponential synchronization of complex dynamical networks (CDNs) under improved nonfragile sampled-data event-triggered control (INFSDETC) is investigated in this study. A meaningful yet challenging issue is solved, namely, it can adjust the triggering mode and the triggering frequency to adapt to more situations in the event-triggered scheme in which it is able to adjust the triggering condition exponentially and linearly, and dynamically adjust the triggering according to the time and state. By using control theory and Lyapunov analysis theory, an improved event-triggered controller was constructed for more intelligent control and to ensure exponential synchronization for CDNs. Lastly, significant numerical simulation examples are developed to show the usefulness and the performance of the proposed methods. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

14 pages, 435 KiB

Open AccessArticle

Improved Stability Criteria for Delayed Neural Networks via a Relaxed Delay-Product-Type Lapunov–Krasovskii Functional

by Shuoting Wang, Kaibo Shi and Jin Yang

Mathematics 2022, 10(15), 2768; https://doi.org/10.3390/math10152768 - 04 Aug 2022

Cited by 3 | Viewed by 984

Abstract

In this paper, the asymptotic stability problem of neural networks with time-varying delays is investigated. First, a new sufficient and necessary condition on a general polynomial inequality was developed. Then, a novel augmented Lyapunov–Krasovskii functional (LKF) was constructed, which efficiently introduces some new [...] Read more.

In this paper, the asymptotic stability problem of neural networks with time-varying delays is investigated. First, a new sufficient and necessary condition on a general polynomial inequality was developed. Then, a novel augmented Lyapunov–Krasovskii functional (LKF) was constructed, which efficiently introduces some new terms related to the previous information of neuron activation function. Furthermore, based on the suitable LKF and the stated negative condition of the general polynomial, two criteria with less conservatism were derived in the form of linear matrix inequalities. Finally, two numerical examples were carried out to confirm the superiority of the proposed criteria, and a larger allowable upper bound of delays was achieved. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

18 pages, 1420 KiB

Open AccessArticle

Impulsive Stabilization on Hyper-Chaotic Financial System under Neumann Boundary

by Xinggui Li, Ruofeng Rao and Xinsong Yang

Mathematics 2022, 10(11), 1866; https://doi.org/10.3390/math10111866 - 30 May 2022

Cited by 4 | Viewed by 1114

Abstract

This paper proposes a novel technique to obtain sufficient conditions for the existence and stabilization of positive solutions for a kind of hyper-chaotic financial model. Since some important economic indexes are heavily related to region, the authors consider a nonlinear chaotic financial system [...] Read more.

This paper proposes a novel technique to obtain sufficient conditions for the existence and stabilization of positive solutions for a kind of hyper-chaotic financial model. Since some important economic indexes are heavily related to region, the authors consider a nonlinear chaotic financial system with diffusion, which leads to some mathematical difficulties in dealing with the infinite-dimension characteristic. In order to overcome these difficulties, novel analysis techniques have to be proposed on the basis of Laplacian semigroup and impulsive control. Sufficient conditions are provided for existence and global exponential stabilization of positive solution for the system. It is interesting to discover that the impulse strength can be larger than 1 in the newly obtained stability criterion. Numerical simulations show the effectiveness of theoretical analysis. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

16 pages, 697 KiB

Open AccessEditor’s ChoiceArticle

Fixed-Time Synchronization for Fuzzy-Based Impulsive Complex Networks

by Lu Pang, Cheng Hu, Juan Yu and Haijun Jiang

Mathematics 2022, 10(9), 1533; https://doi.org/10.3390/math10091533 - 03 May 2022

Cited by 6 | Viewed by 1563

Abstract

This paper mainly deals with the issue of fixed-time synchronization of fuzzy-based impulsive complex networks. By developing fixed-time stability of impulsive systems and proposing a T-S fuzzy control strategy with pure power-law form, some simple criteria are acquired to achieve fixed-time synchronization of [...] Read more.

This paper mainly deals with the issue of fixed-time synchronization of fuzzy-based impulsive complex networks. By developing fixed-time stability of impulsive systems and proposing a T-S fuzzy control strategy with pure power-law form, some simple criteria are acquired to achieve fixed-time synchronization of fuzzy-based impulsive complex networks and the estimation of the synchronized time is given. Ultimately, the presented control scheme and synchronization criteria are verified by numerical simulation. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

13 pages, 941 KiB

Open AccessArticle

Polynomial Noises for Nonlinear Systems with Nonlinear Impulses and Time-Varying Delays

by Lichao Feng, Qiaona Wang, Chunyan Zhang and Dianxuan Gong

Mathematics 2022, 10(9), 1525; https://doi.org/10.3390/math10091525 - 02 May 2022

Cited by 1 | Viewed by 958

Abstract

It is known that random noises have a significant impact on differential systems. Recently, the influences of random noises for impulsive systems have been started. Nevertheless, the existing references on this issue ignore the significant phenomena of nonlinear impulses and time-varying delays. Therefore, [...] Read more.

It is known that random noises have a significant impact on differential systems. Recently, the influences of random noises for impulsive systems have been started. Nevertheless, the existing references on this issue ignore the significant phenomena of nonlinear impulses and time-varying delays. Therefore, we see the necessity to study the influences of random noises for impulsive systems with the above two factors. Stimulated by the above, a polynomial random noise is introduced to suppress the potential explosive behavior of the nonlinear impulsive differential system with time-varying delay. Fortunately, the stochastically controlled impulsive delay differential system admits a unique global solution, is bounded, and grows at most in the polynomial form. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

27 pages, 1549 KiB

Open AccessArticle

Tracking Control for Triple-Integrator and Quintuple-Integrator Systems with Single Input Using Zhang Neural Network with Time Delay Caused by Backward Finite-Divided Difference Formulas for Multiple-Order Derivatives

by Pengfei Guo and Yunong Zhang

Mathematics 2022, 10(9), 1440; https://doi.org/10.3390/math10091440 - 24 Apr 2022

Cited by 4 | Viewed by 1567

Abstract

Tracking control for multiple-integrator systems is regarded as a fundamental problem associated with nonlinear dynamic systems in the physical and mathematical sciences, with many applications in engineering fields. In this paper, we adopt the Zhang neural network method to solve this nonlinear dynamic [...] Read more.

Tracking control for multiple-integrator systems is regarded as a fundamental problem associated with nonlinear dynamic systems in the physical and mathematical sciences, with many applications in engineering fields. In this paper, we adopt the Zhang neural network method to solve this nonlinear dynamic problem. In addition, in order to adapt to the requirements of real-world hardware implementations with higher-order precision for this problem, the multiple-order derivatives in the Zhang neural network method are estimated using backward finite-divided difference formulas with quadratic-order precision, thus producing time delays. As such, we name the proposed method the Zhang neural network method with time delay. Moreover, we present five theorems to describe the convergence property of the Zhang neural network method without time delay and the quadratic-order error pattern of the Zhang neural network method with time delay derived from the backward finite-divided difference formulas with quadratic-order precision, which specifically demonstrate the effect of the time delay. Finally, tracking controllers with quadratic-order precision for multiple-integrator systems are constructed using the Zhang neural network method with time delay, and two numerical experiments are presented to substantiate the theoretical results for the Zhang neural network methods with and without time delay. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures

Figure 1

15 pages, 483 KiB

Open AccessArticle

Event-Triggered State Estimation for Fractional-Order Neural Networks

by Bingrui Xu and Bing Li

Mathematics 2022, 10(3), 325; https://doi.org/10.3390/math10030325 - 21 Jan 2022

Cited by 10 | Viewed by 1950

Abstract

This paper is concerned with the problem of event-triggered state estimation for a class of fractional-order neural networks. An event-triggering strategy is proposed to reduce the transmission frequency of the output measurement signals with guaranteed state estimation performance requirements. Based on the Lyapunov [...] Read more.

This paper is concerned with the problem of event-triggered state estimation for a class of fractional-order neural networks. An event-triggering strategy is proposed to reduce the transmission frequency of the output measurement signals with guaranteed state estimation performance requirements. Based on the Lyapunov method and properties of fractional-order calculus, a sufficient criterion is established for deriving the Mittag–Leffler stability of the estimation error system. By making full use of the properties of Caputo operator and Mittag–Leffler function, the evolution dynamics of measured error is analyzed so as to exclude the unexpected Zeno phenomenon in the event-triggering strategy. Finally, two numerical examples and simulations are provided to show the effectiveness of the theoretical results. Full article

(This article belongs to the Special Issue Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value)

► Show Figures