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Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science...
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Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (31 August 2023) | Viewed by 29735

Special Issue Editor

Prof. Dr. Quanxin Zhu

grade E-Mail Website
Guest Editor
School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
Interests: Markov processes; stochastic differential equations; stochastic partial differential equations; stochastic stability; stochastic neural networks
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Open Mathematics is a challenging notion for theoretical modelling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, because highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods.

The aim of this Special Issue is to highlight papers that show dynamics, control, optimization, and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. 

Potential topics include, but are not limited to, the following:

  • Stability analysis of discrete and continuous dynamical systems;
  • Nonlinear dynamics in biologically complex systems;
  • Stability and stabilization of stochastic systems;
  • Mathematical models in statistics and probability;
  • Synchronization of oscillators and chaotic systems;
  • Optimization methods of complex systems;
  • Reliability modelling and system optimization;
  • Computation and control over networked systems.

Prof. Dr. Quanxin Zhu
Guest Editor

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Related Special Issue

Published Papers (22 papers)

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Research

17 pages, 482 KiB  
Article
A General Iterative Procedure for Solving Nonsmooth Constrained Generalized Equations
by Wei Ouyang and Kui Mei
Mathematics 2023, 11(22), 4577; https://doi.org/10.3390/math11224577 - 8 Nov 2023
Viewed by 850
Abstract
In this paper, we concentrate on an abstract iterative procedure for solving nonsmooth constrained generalized equations. This procedure employs both the property of weak point-based approximation and the approach of searching for a feasible inexact projection on the constrained set. Utilizing the contraction [...] Read more.
In this paper, we concentrate on an abstract iterative procedure for solving nonsmooth constrained generalized equations. This procedure employs both the property of weak point-based approximation and the approach of searching for a feasible inexact projection on the constrained set. Utilizing the contraction mapping principle, we establish higher order local convergence of the proposed method under the assumption of metric regularity property which ensures that the iterative procedure generates a sequence converging to a solution of the constrained generalized equation. Under strong metric regularity assumptions, we obtain that each sequence generated by this procedure converges to a solution. Furthermore, a restricted version of the proposed method is considered, for which we establish the desired convergence for each iterative sequence without a strong metric subregularity condition. The obtained results are new even for generalized equations without a constraint set. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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15 pages, 586 KiB  
Article
Convex Fault Diagnosis of a Three-Degree-of-Freedom Mechanical Crane
by Julio Guzmán-Rabasa, Francisco Rodríguez, Guillermo Valencia-Palomo, Ildeberto Santos-Ruiz, Samuel Gómez-Peñate and Francisco-Ronay López-Estrada
Mathematics 2023, 11(20), 4258; https://doi.org/10.3390/math11204258 - 12 Oct 2023
Viewed by 1072
Abstract
This paper presents a fault detection and estimation method based on a proportional-integral observer applied to a three-degree-of-freedom mechanical crane. Faults are common in this system and can provoke oscillations that generate a loss of performance and stability. A convex linear parameter varying [...] Read more.
This paper presents a fault detection and estimation method based on a proportional-integral observer applied to a three-degree-of-freedom mechanical crane. Faults are common in this system and can provoke oscillations that generate a loss of performance and stability. A convex linear parameter varying approach is proposed to stabilize the crane and detect and isolate actuator faults to guarantee the crane’s performance. The linear matrix inequalities obtained from candidate Lyapunov functions give sufficient conditions to guarantee the fault estimation method. Finally, numerical simulations are proposed to illustrate the method’s performance and applicability. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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18 pages, 377 KiB  
Article
α-Synchronization of a Class of Unbounded Delayed Inertial Cohen–Grossberg Neural Networks with Delayed Impulses
by Fengjiao Zhang, Yinfang Song and Chao Wang
Mathematics 2023, 11(19), 4096; https://doi.org/10.3390/math11194096 - 27 Sep 2023
Viewed by 999
Abstract
As an essential dynamic behavior, the synchronization of inertial Cohen–Grossberg neural networks (ICGNNs) has received considerable attention due to its successful applications in neural cryptography, public channel cryptography, security communications, and image encryption. In this article, the α-synchronization of a class of [...] Read more.
As an essential dynamic behavior, the synchronization of inertial Cohen–Grossberg neural networks (ICGNNs) has received considerable attention due to its successful applications in neural cryptography, public channel cryptography, security communications, and image encryption. In this article, the α-synchronization of a class of non-autonomous unbounded delayed inertial Cohen–Gossberg neural networks with delayed impulses is investigated. Firstly, several non-autonomous impulsive differential inequalities are established, where unbounded delays, delayed impulses, and time-variable coefficients are incorporated. Subsequently, based on the proposed impulsive differential inequalities and Lyapunov function approach, the feedback controllers are designed, and some criteria for α-synchronization are provided. Finally, the validity of the presented theoretical findings is demonstrated by two specific examples. It is shown that delayed impulses can be viewed as perturbations or stabilizing sources for non-autonomous ICGNNs. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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32 pages, 1083 KiB  
Article
Synchronization of Takagi–Sugeno Fuzzy Time-Delayed Stochastic Bidirectional Associative Memory Neural Networks Driven by Brownian Motion in Pre-Assigned Settling Time
by Chengqiang Wang, Xiangqing Zhao, Can Wang and Zhiwei Lv
Mathematics 2023, 11(17), 3697; https://doi.org/10.3390/math11173697 - 28 Aug 2023
Cited by 2 | Viewed by 853
Abstract
We are devoted, in this paper, to the study of the pre-assigned-time drive-response synchronization problem for a class of Takagi–Sugeno fuzzy logic-based stochastic bidirectional associative memory neural networks, driven by Brownian motion, with continuous-time delay and (finitely and infinitely) distributed time delay. To [...] Read more.
We are devoted, in this paper, to the study of the pre-assigned-time drive-response synchronization problem for a class of Takagi–Sugeno fuzzy logic-based stochastic bidirectional associative memory neural networks, driven by Brownian motion, with continuous-time delay and (finitely and infinitely) distributed time delay. To achieve the drive-response synchronization between the neural network systems, concerned in this paper, and the corresponding response neural network systems (identical to our concerned neural network systems), we bring forward, based on the structural properties, a class of control strategies. By meticulously coining an elaborate Lyapunov–Krasovskii functional, we prove a criterion guaranteeing the desired pre-assigned-time drive-response synchronizability: For any given positive time instant, some of our designed controls make sure that our concerned neural network systems and the corresponding response neural network systems achieve synchronization, with the settling times not exceeding the pre-assigned positive time instant. In addition, we equip our theoretical studies with a numerical example, to illustrate that the synchronization controls designed in this paper are indeed effective. Our concerned neural network systems incorporate several types of time delays simultaneously, in particular, they have a continuous-time delay in their leakage terms, are based on Takagi–Sugeno fuzzy logic, and can be synchronized before any pre-given finite-time instant by the suggested control; therefore, our theoretical results in this paper have wide potential applications in the real world. The conservatism is reduced by introducing parameters in our designed Lyapunov–Krasovskii functional and synchronization control. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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14 pages, 307 KiB  
Article
Positive Solutions to the Discrete Boundary Value Problem of the Kirchhoff Type
by Bahua Lin and Zhan Zhou
Mathematics 2023, 11(16), 3588; https://doi.org/10.3390/math11163588 - 19 Aug 2023
Viewed by 1027
Abstract
The paper aims to study a discrete boundary value problem of the Kirchhoff type based on the critical point theory and the strong maximum principle. Compared to the existing literature, the existence and multiplicity of positive solutions to the problem are considered according [...] Read more.
The paper aims to study a discrete boundary value problem of the Kirchhoff type based on the critical point theory and the strong maximum principle. Compared to the existing literature, the existence and multiplicity of positive solutions to the problem are considered according to the behavior of the nonlinear term f in some points between the zero and positive infinity, which is a new attempt. Under different assumptions of the nonlinear term f, we obtain the determined open intervals of the parameter λ, such that the problem has at least three positive solutions or at least two positive solutions in different intervals. In the end, two concrete examples are used to illustrate our main conclusions. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
9 pages, 271 KiB  
Article
Kato Chaos in Linear Dynamics
by Lixin Jiao, Lidong Wang and Heyong Wang
Mathematics 2023, 11(16), 3540; https://doi.org/10.3390/math11163540 - 16 Aug 2023
Viewed by 1103
Abstract
This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators T¯. The main conclusions are as follows: (1) [...] Read more.
This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators T¯. The main conclusions are as follows: (1) If a linear operator is accessible, then the collection of vectors whose orbit has a subsequence converging to zero is a residual set. (2) For a continuous linear operator defined on Fréchet space, Kato chaos is equivalent to dense Li–Yorke chaos. (3) Kato chaos is preserved under the iteration of linear operators. (4) A sufficient condition is obtained under which the Kato chaos for linear operator T and its induced operators T¯ are equivalent. (5) A continuous linear operator is sensitive if and only if its inducing operator T¯ is sensitive. It should be noted that this equivalence does not hold for nonlinear dynamics. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
11 pages, 1262 KiB  
Article
Dynamics and Embedded Solitons of Stochastic Quadratic and Cubic Nonlinear Susceptibilities with Multiplicative White Noise in the Itô Sense
by Zhao Li and Chen Peng
Mathematics 2023, 11(14), 3185; https://doi.org/10.3390/math11143185 - 20 Jul 2023
Cited by 11 | Viewed by 884
Abstract
The main purpose of this paper is to study the dynamics and embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities in the Itô sense, which can further help researchers understand the propagation of soliton nonlinear systems. Firstly, a two-dimensional dynamics system and [...] Read more.
The main purpose of this paper is to study the dynamics and embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities in the Itô sense, which can further help researchers understand the propagation of soliton nonlinear systems. Firstly, a two-dimensional dynamics system and its perturbation system are obtained by using a traveling wave transformation. Secondly, the phase portraits of the two-dimensional dynamics system are plotted. Furthermore, the chaotic behavior, two-dimensional phase portraits, three-dimensional phase portraits and sensitivity of the perturbation system are analyzed via Maple software. Finally, the embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities are obtained. Moreover, three-dimensional and two-dimensional solitons of stochastic quadratic and cubic nonlinear susceptibilities are plotted. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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19 pages, 829 KiB  
Article
Almost Sure Exponential Stability of Uncertain Stochastic Hopfield Neural Networks Based on Subadditive Measures
by Zhifu Jia and Cunlin Li
Mathematics 2023, 11(14), 3110; https://doi.org/10.3390/math11143110 - 14 Jul 2023
Cited by 9 | Viewed by 1041
Abstract
For this paper, we consider the almost sure exponential stability of uncertain stochastic Hopfield neural networks based on subadditive measures. Firstly, we deduce two corollaries, using the Itô–Liu formula. Then, we introduce the concept of almost sure exponential stability for uncertain stochastic Hopfield [...] Read more.
For this paper, we consider the almost sure exponential stability of uncertain stochastic Hopfield neural networks based on subadditive measures. Firstly, we deduce two corollaries, using the Itô–Liu formula. Then, we introduce the concept of almost sure exponential stability for uncertain stochastic Hopfield neural networks. Next, we investigate the almost sure exponential stability of uncertain stochastic Hopfield neural networks, using the Lyapunov method, Liu inequality, the Liu lemma, and exponential martingale inequality. In addition, we prove two sufficient conditions for almost sure exponential stability. Furthermore, we consider stabilization with linear uncertain stochastic perturbation and present some exceptional examples. Finally, our paper provides our conclusion. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
17 pages, 928 KiB  
Article
Global Synchronization of Fractional-Order Multi-Delay Coupled Neural Networks with Multi-Link Complicated Structures via Hybrid Impulsive Control
by Hongguang Fan, Yue Rao, Kaibo Shi and Hui Wen
Mathematics 2023, 11(14), 3051; https://doi.org/10.3390/math11143051 - 10 Jul 2023
Cited by 6 | Viewed by 1081
Abstract
This study discusses the global asymptotical synchronization of fractional-order multi-delay coupled neural networks (FMCNNs) via hybrid control schemes. In addition to internal delays and different coupling delays, more importantly, multi-link complicated structures are introduced into our model. Unlike most existing works, the synchronization [...] Read more.
This study discusses the global asymptotical synchronization of fractional-order multi-delay coupled neural networks (FMCNNs) via hybrid control schemes. In addition to internal delays and different coupling delays, more importantly, multi-link complicated structures are introduced into our model. Unlike most existing works, the synchronization target is not the special solution of an isolated node, and a more universally accepted synchronization goal involving the average neuron states is introduced. A generalized multi-delay impulsive comparison principle with fractional order is given to solve the difficulties resulting from different delays and multi-link structures. To reduce control costs, a pinned node strategy based on the principle of statistical sorting is provided, and then a new hybrid impulsive pinning control method is established. Based on fractional-order impulsive inequalities, Laplace transforms, and fractional order stability theory, novel synchronization criteria are derived to guarantee the asymptotical synchronization of the considered FMCNN. The derived theoretical results can effectively extend the existing achievements for fractional-order neural networks with a multi-link nature. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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14 pages, 370 KiB  
Article
A Dual Rumor Spreading Model with Consideration of Fans versus Ordinary People
by Hongying Xiao, Zhaofeng Li, Yuanyuan Zhang, Hong Lin and Yuxiao Zhao
Mathematics 2023, 11(13), 2958; https://doi.org/10.3390/math11132958 - 3 Jul 2023
Cited by 1 | Viewed by 1079
Abstract
The spread of rumors in online social networks (OSNs) has caused a serious threat to the normal social order. In order to describe the rumor-spreading dynamics in OSNs during emergencies, a novel model with consideration of fans versus ordinary people is proposed in [...] Read more.
The spread of rumors in online social networks (OSNs) has caused a serious threat to the normal social order. In order to describe the rumor-spreading dynamics in OSNs during emergencies, a novel model with consideration of fans versus ordinary people is proposed in this paper. In contrast to previous studies, we consider the case that two rumors exist simultaneously. It is assumed that one is an entertainment rumor that fans care about, and the other is a common rumor. First, we derive the mean-field equations that describe the dynamics of this dual rumor propagation model and obtain the threshold parameter. Secondly, after finding the necessary and sufficient conditions for the existence of equilibriums, we examine the equilibrium’s local and global stability. Finally, simulations are used to explain how various parameters affect the process of spreading rumors. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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29 pages, 3440 KiB  
Article
Multiple-Frequency Force Estimation of Controlled Vibrating Systems with Generalized Nonlinear Stiffness
by Francisco Beltran-Carbajal, Juan Eduardo Esquivel-Cruz, Hugo Yañez-Badillo, Ivan de Jesus Rivas-Cambero, David Sotelo and Carlos Sotelo
Mathematics 2023, 11(13), 2838; https://doi.org/10.3390/math11132838 - 24 Jun 2023
Cited by 2 | Viewed by 1226
Abstract
An on-line estimation technique of multiple-frequency oscillatory forces combined with the Hilbert–Huang transform for an important class of actively controlled, forced vibrating mechanical systems with nonlinear stiffness forces is proposed. Polynomial parametric nonlinearities are incorporated in the significantly perturbed vibrating system dynamics. This [...] Read more.
An on-line estimation technique of multiple-frequency oscillatory forces combined with the Hilbert–Huang transform for an important class of actively controlled, forced vibrating mechanical systems with nonlinear stiffness forces is proposed. Polynomial parametric nonlinearities are incorporated in the significantly perturbed vibrating system dynamics. This class of nonlinear vibrating systems can exhibit harmful large-amplitude vibrations, which are inadmissible in many engineering applications. Disturbing oscillations can be also provoked due to interactions of the primary mechanical system to be actively protected against dangerous vibrations with other forced uncertain multidegree-of-freedom nonlinear vibrating systems. Taylor’s series expansion to dynamically model uncertain vibrating forces into a small time window for real-time estimation purposes is employed. Intrinsic mode functions of multiple-frequency vibrating forces can be then obtained by the Hilbert-Huang transform. Uncertain instantaneous frequencies and amplitudes of disturbing oscillations can be directly computed in temporal space. An active vibration control scheme for efficient and robust tracking of prescribed motion reference profiles based on multiple frequency force estimation is introduced as well. The presented closed-loop on-line estimation technique can be extended for other classes of nonlinear oscillatory systems. Analytical, experimental and numerical results to prove the estimation effectiveness are presented. Numerical results show reasonable estimation errors of less than 2%. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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13 pages, 323 KiB  
Article
Positive Periodic Solution for Neutral-Type Integral Differential Equation Arising in Epidemic Model
by Qing Yang, Xiaojing Wang, Xiwang Cheng, Bo Du and Yuxiao Zhao
Mathematics 2023, 11(12), 2701; https://doi.org/10.3390/math11122701 - 14 Jun 2023
Cited by 4 | Viewed by 942
Abstract
This paper is devoted to investigating a class of neutral-type integral differential equations arising in an epidemic model. By using Mawhin’s continuation theorem and the properties of neutral-type operators, we obtain the existence conditions for positive periodic solutions of the considered neutral-type integral [...] Read more.
This paper is devoted to investigating a class of neutral-type integral differential equations arising in an epidemic model. By using Mawhin’s continuation theorem and the properties of neutral-type operators, we obtain the existence conditions for positive periodic solutions of the considered neutral-type integral differential equation. Compared with previous results, the existence conditions in this paper are less restricted, thus extending the results of the existing literature. Finally, two examples are given to show the effectiveness and merits of the main results of this paper. Our results can be used to obtain the existence of a positive periodic solution to the corresponding non-neutral-type integral differential equation. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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14 pages, 417 KiB  
Article
Exponential Stability of a Class of Neutral Inertial Neural Networks with Multi-Proportional Delays and Leakage Delays
by Chao Wang, Yinfang Song, Fengjiao Zhang and Yuxiao Zhao
Mathematics 2023, 11(12), 2596; https://doi.org/10.3390/math11122596 - 6 Jun 2023
Cited by 8 | Viewed by 1043
Abstract
This paper investigates the exponential stability of a class of neutral inertial neural networks with multi-proportional delays and leakage delays. By utilizing the Lyapunov stability theory, the approach of parametric variation, and the differential inequality technique, some criteria are acquired that can guarantee [...] Read more.
This paper investigates the exponential stability of a class of neutral inertial neural networks with multi-proportional delays and leakage delays. By utilizing the Lyapunov stability theory, the approach of parametric variation, and the differential inequality technique, some criteria are acquired that can guarantee that all solutions of the addressed system converge exponentially to the equilibrium point. In particular, the neutral term, multi-proportional delays, and leakage delays are incorporated simultaneously, resulting in a more general model, and the findings are novel and refine the previous works. Finally, one example is provided to indicate that the dynamic behavior is consistent with the theoretical analysis. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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16 pages, 738 KiB  
Article
Mean-Square Stability of Uncertain Delayed Stochastic Systems Driven by G-Brownian Motion
by Zhengqi Ma, Shoucheng Yuan, Kexin Meng and Shuli Mei
Mathematics 2023, 11(10), 2405; https://doi.org/10.3390/math11102405 - 22 May 2023
Cited by 14 | Viewed by 1382
Abstract
This paper investigates the mean-square stability of uncertain time-delay stochastic systems driven by G-Brownian motion, which are commonly referred to as G-SDDEs. To derive a new set of sufficient stability conditions, we employ the linear matrix inequality (LMI) method and construct a Lyapunov–Krasovskii [...] Read more.
This paper investigates the mean-square stability of uncertain time-delay stochastic systems driven by G-Brownian motion, which are commonly referred to as G-SDDEs. To derive a new set of sufficient stability conditions, we employ the linear matrix inequality (LMI) method and construct a Lyapunov–Krasovskii function under the constraint of uncertainty bounds. The resulting sufficient condition does not require any specific assumptions on the G-function, making it more practical. Additionally, we provide numerical examples to demonstrate the validity and effectiveness of the proposed approach. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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26 pages, 747 KiB  
Article
Neural Network Trajectory Tracking Control on Electromagnetic Suspension Systems
by Francisco Beltran-Carbajal, Hugo Yañez-Badillo, Ruben Tapia-Olvera, Julio C. Rosas-Caro, Carlos Sotelo and David Sotelo
Mathematics 2023, 11(10), 2272; https://doi.org/10.3390/math11102272 - 12 May 2023
Cited by 4 | Viewed by 1584
Abstract
A new adaptive-like neural control strategy for motion reference trajectory tracking for a nonlinear electromagnetic suspension dynamic system is introduced. Artificial neural networks, differential flatness and sliding modes are strategically integrated in the presented adaptive neural network control design approach. The robustness and [...] Read more.
A new adaptive-like neural control strategy for motion reference trajectory tracking for a nonlinear electromagnetic suspension dynamic system is introduced. Artificial neural networks, differential flatness and sliding modes are strategically integrated in the presented adaptive neural network control design approach. The robustness and efficiency of the magnetic suspension control system on desired smooth position reference profile tracking can be improved in this fashion. A single levitation control parameter is tuned on-line from a neural adaptive perspective by using information of the reference trajectory tracking error signal only. The sliding mode discontinuous control action is approximated by a neural network-based adaptive continuous control function. Control design is firstly developed from theoretical modelling of the nonlinear physical system. Next, dependency on theoretical modelling of the nonlinear dynamic system is substantially reduced by integrating B-spline neural networks and sliding modes in the electromagnetic levitation control technique. On-line accurate estimation of uncertainty, unmeasured external disturbances and uncertain nonlinearities are conveniently evaded. The effective performance of the robust trajectory tracking levitation control approach is depicted for multiple simulation operating scenarios. The capability of active disturbance suppression is furthermore evidenced. The presented B-spline neural network trajectory tracking control design approach based on sliding modes and differential flatness can be extended to other controllable complex uncertain nonlinear dynamic systems where internal and external disturbances represent a relevant issue. Computer simulations and analytical results demonstrate the effective performance of the new adaptive neural control method. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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11 pages, 303 KiB  
Article
Generalized Halanay Inequalities and Relative Application to Time-Delay Dynamical Systems
by Chunsheng Wang, Xiangdong Liu, Feng Jiao, Hong Mai, Han Chen and Runpeng Lin
Mathematics 2023, 11(8), 1940; https://doi.org/10.3390/math11081940 - 20 Apr 2023
Cited by 9 | Viewed by 1230
Abstract
A class of generalized Halanay inequalities is studied via the Banach fixed point method and comparison principle. The conditions to ensure the boundedness and stability of the zero solution are obtained in this study. This research provides a new approach to the study [...] Read more.
A class of generalized Halanay inequalities is studied via the Banach fixed point method and comparison principle. The conditions to ensure the boundedness and stability of the zero solution are obtained in this study. This research provides a new approach to the study of the boundedness and stability of Halanay inequality. Numerical examples and simulation results verify the validity and superiority of the conclusions obtained in this study. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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11 pages, 275 KiB  
Article
Positive Periodic Solution for Pipe/Tank Flow Configurations with Friction
by Haiqing Du, Xiaojing Wang and Bo Du
Mathematics 2023, 11(8), 1789; https://doi.org/10.3390/math11081789 - 9 Apr 2023
Viewed by 1060
Abstract
In this article, we study a periodic boundary value problem related to valveless pumping. The valveless pumping is described by the unidirectional flow of liquid in a system. We establish some conditions for globally asymptotic stability and the existence of a positive periodic [...] Read more.
In this article, we study a periodic boundary value problem related to valveless pumping. The valveless pumping is described by the unidirectional flow of liquid in a system. We establish some conditions for globally asymptotic stability and the existence of a positive periodic solution to the considered equation. Finally, a numerical example shows that the theoretical results in this paper are feasible. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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18 pages, 446 KiB  
Article
Hybrid Impulsive Pinning Control for Mean Square Synchronization of Uncertain Multi-Link Complex Networks with Stochastic Characteristics and Hybrid Delays
by Yong Tang, Lang Zhou, Jiahui Tang, Yue Rao, Hongguang Fan and Jihong Zhu
Mathematics 2023, 11(7), 1697; https://doi.org/10.3390/math11071697 - 2 Apr 2023
Cited by 25 | Viewed by 1841
Abstract
This study explores the synchronization issue for uncertain multi-link complex networks incorporating stochastic characteristics and hybrid delays. Unlike previous works, internal delays, coupling delays, and stochastic delays considered in our model change over time; meanwhile, the impulse strength and position change with time [...] Read more.
This study explores the synchronization issue for uncertain multi-link complex networks incorporating stochastic characteristics and hybrid delays. Unlike previous works, internal delays, coupling delays, and stochastic delays considered in our model change over time; meanwhile, the impulse strength and position change with time evolution. To actualize network synchronization, a strategy called hybrid impulsive pinning control is applied, which combines the virtue of impulsive control and pinning control as well as two categories of impulses (i.e., synchronization and desynchronization). By decomposing the complicated topological structures into diagonal items and off-diagonal items, multiple nonlinear coupling terms are linearly decomposed in the process of theoretical analysis. Combining inequality technology and matrix decomposition theory, several novel synchronization criteria have been gained to ensure synchronization for the concerning multi-link model. The criteria get in touch with the uncertain strengths, coupling strengths, hybrid impulse strengths, delay sizes, impulsive intervals, and network topologies. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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18 pages, 336 KiB  
Article
On a Method for Optimizing Controlled Polynomial Systems with Constraints
by Alexander Buldaev and Dmitry Trunin
Mathematics 2023, 11(7), 1695; https://doi.org/10.3390/math11071695 - 2 Apr 2023
Viewed by 1315
Abstract
A new optimization approach is considered in the class of polynomial in-state optimal control problems with constraints based on nonlocal control improvement conditions, which are constructed in the form of special fixed-point problems in the control space. The proposed method of successive approximations [...] Read more.
A new optimization approach is considered in the class of polynomial in-state optimal control problems with constraints based on nonlocal control improvement conditions, which are constructed in the form of special fixed-point problems in the control space. The proposed method of successive approximations of control retains all constraints at each iteration and does not use the operation of parametric variation of control at each iteration, in contrast to known gradient methods. In addition, the initial approximation of the iterative process may not satisfy the constraints, which is a significant factor in increasing the efficiency of the approach. The comparative efficiency of the proposed method of fixed points in the considered class of problems is illustrated in a model example. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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10 pages, 949 KiB  
Article
Stability Analysis for a Class of Stochastic Differential Equations with Impulses
by Mingli Xia, Linna Liu, Jianyin Fang and Yicheng Zhang
Mathematics 2023, 11(6), 1541; https://doi.org/10.3390/math11061541 - 22 Mar 2023
Cited by 47 | Viewed by 2965
Abstract
This paper is concerned with the problem of asymptotic stability for a class of stochastic differential equations with impulsive effects. A sufficient criterion on asymptotic stability is derived for such impulsive stochastic differential equations via Lyapunov stability theory, bounded difference condition and martingale [...] Read more.
This paper is concerned with the problem of asymptotic stability for a class of stochastic differential equations with impulsive effects. A sufficient criterion on asymptotic stability is derived for such impulsive stochastic differential equations via Lyapunov stability theory, bounded difference condition and martingale convergence theorem. The results show that the impulses can facilitate the stability of the stochastic differential equations when the original system is not stable. Finally, the feasibility of our results is confirmed by two numerical examples and their simulations. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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17 pages, 2461 KiB  
Article
Periodically Intermittent Control of Memristor-Based Hyper-Chaotic Bao-like System
by Kun Li, Rongfeng Li, Longzhou Cao, Yuming Feng and Babatunde Oluwaseun Onasanya
Mathematics 2023, 11(5), 1264; https://doi.org/10.3390/math11051264 - 6 Mar 2023
Cited by 23 | Viewed by 2010
Abstract
In this paper, based on a three-dimensional Bao system, a memristor-based hyper-chaotic Bao-like system is successfully constructed, and a simulated equivalent circuit is designed, which is used to verify the chaotic behaviors of the system. Meanwhile, a control method called periodically intermittent control [...] Read more.
In this paper, based on a three-dimensional Bao system, a memristor-based hyper-chaotic Bao-like system is successfully constructed, and a simulated equivalent circuit is designed, which is used to verify the chaotic behaviors of the system. Meanwhile, a control method called periodically intermittent control with variable control width is proposed. The control width sequence in the proposed method is not only variable, but also monotonically decreasing, and the method can effectively stabilize most existing nonlinear systems. Moreover, the memristor-based hyper-chaotic Bao-like system is controlled by combining the proposed method with the Lyapunov stability principle. Finally, we should that the proposed method can effectively control and stabilize not only the proposed hyper-chaotic system, but also the Chua’s oscillator. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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12 pages, 290 KiB  
Article
Constructions of Goethals–Seidel Sequences by Using k-Partition
by Shuhui Shen and Xiaojun Zhang
Mathematics 2023, 11(2), 294; https://doi.org/10.3390/math11020294 - 6 Jan 2023
Cited by 2 | Viewed by 1441
Abstract
In this paper, we are devoted to finding Goethals–Seidel sequences by using k-partition, and based on the finite Parseval relation, the construction of Goethals–Seidel sequences could be transformed to the construction of the associated polynomials. Three different structures of Goethals–Seidel sequences will [...] Read more.
In this paper, we are devoted to finding Goethals–Seidel sequences by using k-partition, and based on the finite Parseval relation, the construction of Goethals–Seidel sequences could be transformed to the construction of the associated polynomials. Three different structures of Goethals–Seidel sequences will be presented. We first propose a method based on T-matrices directly to obtain a quad of Goethals–Seidel sequences. Next, by introducing the k-partition, we utilize two classes of 8-partitions to obtain a new class of polynomials still remaining the same (anti)symmetrical properties, with which a quad of Goethals–Seidel sequences could be constructed. Moreover, an adoption of the 4-partition together with a quad of four symmetrical sequences can also lead to a quad of Goethals–Seidel sequences. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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