In this article, the practical fixed-time (FxT) control of switched neutral Filippov systems (SNFSs) on networks is considered. The perturbation functions that can be discontinuous, and the neutral logics expressed by the difference operators, are addressed by new approaches. Several novel Lyapunov inequalities with indefinite functions are proposed, where detailed estimations of the settling-time (ST) are obtained by discussing the different values of the exponents of the Lyapunov functions, which can include the existing results. Considering that the states generally cannot converge to the origin accurately under finite-time control in real applications, practical FxT stability lemmas with indefinite functions are established for the first time, in which the bounded condition imposed on the indefinite function is more practical than the previously unbounded ones. By designing the adaptive control strategies, the FxT and practical FxT synchronization control are investigated based on the Lyapunov-Krasovskii functionals (LKFs), which show the delay characteristic via the adaptive update law containing delay values. Notably, the theoretical deficiency arising from the Lyapunov function when studying the FxT stability of real systems with delays by using the FxT stability lemmas with indefinite function is solved in a successful way. Finally, the validity of the main results is verified by numerical simulations on an electrical device containing an LC transmission line.

In this paper, the fixed-time (FxT) and practical FxT consensus of discontinuous multi-agent systems (MASs) via adaptive intermittent control strategy are studied. New intermittent FxT stability lemmas incorporating the indefinite function and the unified exponent condition are established, which can include the existing periodically intermittent stability lemmas and aperiodic intermittent lemmas with negative definite derivatives. Considering that the MASs cannot realize the exact consensus under the inevitable external interferences, intermittent-type practical FxT stability lemmas are further obtained. The intermittent controllers and adaptive intermittent controllers are designed, which demonstrates the improvements over the existing control strategies since the chattering phenomenon and singularity do not appear. Notably, the control gains are time-varying, which are also different from the previous time-invariant ones. Based on the established stability lemmas, the FxT consensus and practical FxT consensus are achieved. Clearly, the practical FxT consensus of discontinuous MASs under the discontinuous-time controllers is achieved for the first time. The simulation confirms the applicability of the proposed methods and shows strong support for our theoretical findings.

In this paper, fixed-time and practical fixed-time control of switched Filippov systems on networks with nondifferential delays are studied. Firstly, by using the comparison principle and intermittent control interval method, intermittent fixed-time stability lemmas with unified exponential conditions are proposed. The estimation of the settling-time shows that the elasticity number in the control interval plays a vital role. In order to solve the unavailability of global information, without using the auxiliary timer function, intermittent practical fixed-time lemmas with unified exponential conditions are established, which can improve the related ones greatly. It is necessary to point out that the residual set is relevant to the constant parameters of the system. The constant parameters of the system can be used to get a needed attraction region. Based on newly established stability lemmas, the fixed-time control and practical fixed-time control of the addressed switched Filippov systems are achieved based on the designed intermittent-type quantized control strategies, which enables the quantized control to be performed in intermittent intervals. Some previous control strategies are improved. Finally, numerical examples and simulations are provided to verify the validity of the obtained results. Note to Practitioners—Intermittent control has become popular recently since it saves more costs and is more practical in engineering. As for the complex networks, it is not enough to use intermittent control only, but also to consider the bandwidth of signal transmission. So, how to enable the quantized control to be performed in intermittent intervals is an interesting question. In addition, the global information of many models is unidentifiable, practical fixed-time control will be more realistic. Is the residual set in practical fixed-time control relevant to the intermittent control interval? These questions puzzled the authors and prompted this study.

This article aims to consider the practical fixed-time (FxT) synchronization control of discontinuous Takagi–Sugeno (T–S) fuzzy complex networks (DTSFCNs). By using the comparison principle and the iterative technique, new completely-intermittent-type FxT stability lemmas are established. The new Lyapunov inequality with a unified exponent condition is proposed, which can reduce the input of redundant system-independent parameters. Some existing results on the FxT stability lemmas are improved. Considering that the states cannot converge to the origin accurately, completely-intermittent-type practical FxT stability lemmas are further investigated, which can be regarded as the first one and the problem of constructing the practical FxT stability lemmas under the completely-intermittent mechanism is solved. From the perspective of saving information resources to the greatest extent, by designing the quantized controllers, the intermittent FxT and practical FxT control of the addressed DTSFCNs are studied. Here, the quantized controller only works in intermittent intervals, which can improve the quantized controllers and intermittent controllers. Finally, numerical simulations are given to verify the main results.

This paper studies the practical fixed-time synchronization control of T-S fuzzy complex networks with different dimensions, a topic for which no prior solutions exist, despite extensive research on the synchronization of T-S fuzzy complex networks with the same dimensions. First of all, under the semi-intermittent framework, by adopting a more relaxed average control rate under the new definition, a novel fixed-time stability lemma with a unified exponential condition is established, which can reduce the input of parameters irrelevant to the system. Considering the important practical application significance of the practical fixed-time stability, two new practical fixed-time stability lemmas are further obtained based on two different algorithms, which can greatly improve the existing ones. By using the transformation of the same dimension, some new results on the practical fixed-time synchronization control of the considered networks are given by designing the intermittent event-triggered controllers with exponential gains, which have smaller trigger frequency and higher trigger efficiency than the traditional event-triggered controllers. In parallel, both static and dynamic event-triggered controls have been designed, and the dynamic function has been stringently proven to be positive via simple conditions. This accomplishment addresses a long-standing issue in the extant literature, where the presence of high-order terms in the dynamic function had hitherto precluded or failed to establish its non-negativity. Finally, the viability of the acquired results is demonstrated through numerical simulations.

In recent years, the synchronization of coupled neural networks (CNNs) has been extensively studied. However, existing results heavily rely on assuming continuous couplings, overlooking the prevalence of intermittent couplings in reality. In this article, we address for the first time the synchronization challenge posed by intermittently CNNs (ICNNs) with coupling delay. To overcome the difficulties arising from intermittent couplings, we put forward a general piecewise delay differential inequality to characterize the dynamics during both coupled intervals and decoupled intervals. Based on the proposed inequality, we establish delay-independent synchronization criteria (DISCs) for ICNNs, enabling them to tackle general coupling delay. Notably, unlike previous studies, the achievement of synchronization in our approach does not rely on external control. Furthermore, for ICNNs that synchronize only under small delays, we formulate non-linear matrix inequality (LMI)-based delay-dependent synchronization criteria (DDSCs) that are computationally efficient and do not require delay differentiability. Finally, we provide illustrative examples to demonstrate our theoretical results.

In this article, the predefined-time (PdT) control of the switched neutral Filippov system (SNFS) is studied. Semi-intermittent-type and completely intermittent-type PdT stability lemmas are initially established. The indefinite function, united exponent parameter and intermittent time condition are adopted. The new completely intermittent-type PdT stability lemma shows that intermittent interval is related to the PdT parameter, and semi-intermittent-type one is not. Such results are new and some previous PdT stability lemmas are improved greatly. As an application, the SNFS is proposed, which is more generalized. Under the semi/completely intermittent time conditions framework, by designing discontinuous control strategy and bounded control strategy, the zero solution of the considered SNFS is PdT stable. For the PdT control, the control input constraint is considered for the first time. Finally, a numerical example and simulations are provided to illustrate the correctness of the main results.

This paper aims to consider the finite-time synchronization of a class of discontinuous fuzzy competitive networks with non-differential leakage delays. Different from the previous competitive networks and leakage-delayed networks, the activation functions are discontinuous and the leakage delays are non-differential. Without using the Lyapunov function methods, the synchronization in a finite-time is investigated based on the comparison principle and inequality techniques, which are used for the first time in the study of the competitive networks and leakage-delayed networks. Moreover, the feedback controller and bounded feedback controller are designed to help achieve the finite-time synchronization. The bounded feedback controller is formulated by the functionarctanand a new inequality lemma is given, which plays a vital role in achieving the synchronization via bounded feedback controller. Although both controllers can help achieve the finite-time synchronization, it can be found that there is a big difference that the value of feedback controller is greatly influenced by system's state variables, but bounded controller is not. Such fact leads to different settling-times. The conclusion is that bounded controller has more advantages in shortening convergence time through numerical simulations.

In this article, finite/fixed-time stabilization and energy consumption estimation of a class of discontinuous fuzzy inertial Cohen-Grossberg neural networks (CGNNs) are considered. The saturation function is utilized to design the control laws, which is different from the previous ones with the sign function since the saturation function is continuous and can reduce the chattering phenomenon. By using the Lyapunov method, comparison principle, and differential inequality theory, the finite/fixed-time stabilization is analyzed by analyzing the state variables inside and outside the unit spherical area, which is different from the direct calculation method using finite/fixed-time stability lemmas. Detailed analyses show the differences in the estimations of the settling time between the initial values inside and outside of the unit spherical area. The energy consumption is also estimated when the finite/fixed-time stabilization is achieved under the designed controller, which is discussed for the first time in the inertial networks. Finally, the validity of the theoretical results is demonstrated by a typical numerical example.

This article focuses on the fixed-time (FxT) synchronization of discontinuous fuzzy competitive neural networks (DFCNNs) with leakage-delays and state-dependent switching. Discontinuous activations, fuzzy terms, leakage delays, and state-dependent switching are considered into the competitive neural networks to increase the generality in practical applications. Before studying the synchronization, as an important theoretical basis, a new FxT stability lemma is established based on the method of nonscaling integral denominator, which is low conservative. Five different settling-time estimates are explicitly given by discussing the range of the exponents. Previous results are included and greatly improved. In order to achieve the FxT synchronization of the addressed DFCNNs, a continuously bounded control scheme is proposed by introducing a saturation function rather than a sign function, which cannot only solve the chattering phenomenon, but also ensure the boundedness of the control input. Some new leakage-delay-dependent synchronization criteria are derived for the DFCNNs. Finally, the validity of the main results is verified by numerical simulations.

This article aims to study the fixed-time (FxT) synchronization of different dimensional Filippov systems. New FxT stability lemmas containing the classical inequality$\dot {V}\leq {-c}_{1}V^{a}-c_{2}V^{b}$proposed by Polyakov are established. Different from the previous FxT stability lemmas in the literature, the proposed one shows the new conclusion that the settling times can be larger or smaller as long as$c_{1}$and$c_{2}$satisfy the certain relationships, which synchronously reveals that the relationships between the control gains can lead to different settling times. Besides, a generalized economical inequality condition$\dot {V} \leq -c_{1}V-c_{2}V^{\upsilon +{\mathrm{ sign}}(V-r)}$is proposed and a novel FxT stability lemma is also established. The complete theoretical proof is given to reveal that$r=1$leads to desired settling time, some previous related results are improved. Based on the new FxT stability lemmas and differential inclusion theory, algebraic inequality conditions are provided to guarantee the FxT synchronization, which reports the first result on the FxT synchronization of different dimensional Filippov systems. Finally, numerical examples are provided to verify the correctness of the main theoretical results.

In this article, a general Filippov system with nondifferentiable and unbounded delays is studied, which is different from the previous delayed Filippov systems. First, based on the differential inclusion theory and Kakutani’s fixed point theorem, the periodicity of solutions is proved and sufficient criteria are derived. Second, fixed-time (FxT) stabilization is studied by designing a new control law with a unified steepness exponent, which is more simpler than the previous ones with two steepness exponents. Since the delays are nondifferentiable and unbounded, rather than constructing the Lyapunov–Krasovskii functions, by constructing a suitable Lyapunov function, the FxT stabilization is analyzed and estimation of the settling-time (ST) is given through analyzing the state variables inside and outside the unit spherical area. The energy consumption is also estimated when the FxT stabilization is achieved under the designed controller. Third, given that smaller ST and lower energy consumption are usually preferred, the optimization problem is further considered. The optimal control parameters are selected based on the normalization approach and maximum principle. Finally, the validity of the theoretical results is demonstrated by a numerical example, where the delay function is composed of the Takagi function and the absolute value function.

In this article, the finite-time (FnT) control of a class of discontinuous memristor-based fuzzy inertial neural networks with proportional delays is discussed. The proposed model is more generalized. Different from the use of the finite/fixed-time stability lemmas in the previous works, by constructing the unit ball and discussing the subordinate relationship between the states and the unit ball, the finite/fixed-time stability are analyzed in detail. This method can avoid the rough thinking that the state variables of the system start from greater than 1. Also, the control constraint is taken into account and a quantized controller is designed, which can not only help achieve the finite/fixed-time stability, but also can prevent the singularity when using the norm forms and chattering when using the sign function in the controllers. In addition, the energy consumptions generated by the quantized controller are estimated in FnT and fixed-time stability processes. The criteria show that the energy consumptions are relevant to the proportional delays, when the proportional coefficient is 1, the energy loss reaches the minimum value. Finally, numerical example and simulation are provided to verify the validity of the obtained results.

This paper aims to study the fixed-time stabilization of a class of delayed discontinuous reaction-diffusion Cohen-Grossberg neural networks. Firstly, by providing some relaxed conditions containing indefinite functions and based on inequality techniques, a new fixed-time stability lemma is given, which can improve the traditional ones. Secondly, based on state-dependent switching laws, the periodic wave solution of the formulated networks is transformed into the periodic solution of ordinary differential system. By utilizing differential inclusions theory and coincidence theorem, the existence of periodic solutions is obtained. Thirdly, based on the new fixed-time stability lemma, the periodic solutions are stabilized at zero in a fixed-time, which is a new topic on reaction-diffusion networks. Moreover, the established criteria are all delay-dependent, which are less conservative than the previous delay-independent ones for ensuring the stabilization of delayed reaction-diffusion networks. Finally, two examples give numerical explanations of the proposed results and highlight the influence of delays.

This article mainly focuses on putting forward new fixed-time (FIXT) stability lemmas of delayed Filippov discontinuous systems (FDSs). By providing the new inequality conditions imposed on the Lyapunov-Krasovskii functions (LKF), novel FIXT stability lemmas are investigated with the help of inequality techniques. The new settling time is also given and its accuracy is improved in comparison with pioneer ones. For the purpose of illustrating the applicability, a class of discontinuous fuzzy neutral-type neural networks (DFNTNNs) is considered, which includes the previous NTNNs. New criteria are derived and detailed FIXT synchronization results have been obtained. Finally, typical examples are carried out to demonstrate the validity of the main results.

When studying the stability of time-delayed discontinuous systems, Lyapunov-Krasovskii functional (LKF) is an essential tool. More relaxed conditions imposed on the LKF are preferred and can take more advantages in real applications. In this article, novel conditions imposed on the LKF are first given which are different from the previous ones. New fixed-time (FXT) stability lemmas are established using some inequality techniques which can greatly extend the pioneers. The new estimations of the settling times (STs) are also obtained. For the purpose of examining the applicability of the new FXT stability lemmas, a class of discontinuous neutral-type neural networks (NTNNs) with proportional delays is formulated which is more generalized than the existing ones. Using differential inclusions theory, set-valued map, and the newly obtained FXT stability lemma, some algebraic FXT stabilization criteria are derived. Finally, examples are given to show the correctness of the established results.

This article proposes fixed-time stability lemmas for the Filippov system via some new inequality approaches. The adopted method no longer needs to integrate the Lyapunov function$V$on the two integral intervals, which is quite different from the existing ones. Some new estimations of the settling times are provided. Also, the steepness exponents of the Lyapunov function$V$in the previous fixed-time stability lemmas are improved. In order to further study the nondifferentiable delayed neural networks modeled by the Filippov system, a class of discontinuous uncertain Cohen–Grossberg neural networks (CGNNs) with mixed delays is formulated and the distributed delays are nondifferentiable, which is more general. Due to the existence of nondifferentiable distributed delays, the existence of the periodic solutions is proved by Kakutani’s fixed-point theorem before considering the stability. By virtue of the obtained fixed-time stability lemmas and the constructed delay-product-type Lyapunov–Krasovskii functional, the fixed-time stabilization is obtained via a no-chattering controller. Clearly, the designed controller does not contain integral terms and delay terms for dealing with the time delays in the closed-loop system, which is more simplified and practical. Finally, two examples help examine the correctness of the main results.

As is known to all that the Lyapunov-Krasovskii functional (LKF) has played a vital role in studying the fixed-time (FXT) stability of the time-delayed Filippov discontinuous systems. When investigating the FXT stability of the time-delayed discontinuous systems, the interesting topic is to provide the more relaxed conditions imposed on the LKF and obtain the more accurate settling-time (ST). This paper aims to investigate the new FXT stability lemma with more relaxed conditions for the time-delayed Filippov discontinuous systems. By using some novel inequality techniques and analytical methods, the new FXT stability lemma is established and more accurate ST is estimated. The traditional discussion about$0 < V < 1$and$V \geq 1$is not required. For the purpose of validation, FXT stabilization of discontinuous inertial networks with mixed time delays is considered. By using differential inclusions theory and the new FXT stability lemma, the delay-dependent FXT stabilization criteria are derived. It is the first time to derive the delay-dependent FXT stabilization criteria for the discontinuous inertial networks. Finally, examples are given to show the effectiveness of the established results.

This article discusses the fixed-time stability (FTS) of a kind of delayed discontinuous system (DS) in Filippov sense. Based on the set-valued map, the FTS analysis of the general solution is first transformed into the zero solution of the differential inclusion. Second, the new criteria of the Lyapunov-Krasovskii functional (LKF) are given and LKF is proved to possess the indefinite derivatives by using the simple integral inequalities. In addition, the FTS of the considered delayed DS is achieved and the new settling time is estimated. Third, to demonstrate the applicability of the new FTS theorems, the FTS control of a class of discontinuous inertial neural networks (DINNs) with time-varying delays is solved. Finally, two numerical examples are given to examine the theoretical results and simulations are also provided to make some illustrations.

This article aims to propose time-delayed and simplified inequality conditions and establish more novel fixed-time (FXT) stability principles of delayed Filippov systems. First, the sign function is used to unify the orders of the Lyapunov–Krasovskii functional, and the time-delayed term is introduced in the inequality conditions, then some new FXT stability theorems are established and new estimations of the settling time involving delayed parameter are given that are absolutely different from the previous ones. The traditional inequality conditions$\dot{V}(z)\leq -aV^\alpha (z)-bV^\beta (z)$is optimized and improved. For the purpose of illustrating the applicability, a class of delayed discontinuous fuzzy Cohen–Grossberg neural networks (DDFCGNNs) is studied. Some new criteria are derived, and the FXT stabilization of the addressed DDFCGNNs is achieved. Finally, a representative example and simulations are provided to examine the correctness.

This paper presents the analysis and intelligent prediction for the displacement of stratum and tunnel lining of Qingdao Metro Line 4 by earth pressure balance (EPB) shield tunnel excavation in complex strata. When the tunnel is excavated in different stratum sections, the tunneling parameters of shield machine are systematically analyzed and compared, and the vertical displacement of the tunnel crown and the horizontal convergence deformation on both sides are investigated. When the tunnel body passes through the soft soil stratum and rock stratum, the curves of the vertical displacement of the stratum surface with time are respectively discussed. A machine learning method for predicting stratum surface deformation induced by shield tunnel excavation in complex strata is developed, where extreme learning machine (ELM), particle swarm optimization (PSO) algorithm and$k$-fold cross-validation method are comprehensively considered. 65 data samples are collected from the field monitoring data of Qingdao Metro Line 4 and each data sample includes seven input values and one output value. The developed PSO-ELM has good prediction performance for stratum surface vertical displacement due to shield tunnel excavation. The case study in this work can provide a practical reference for similar tunneling projects.

In this article, a class of discontinuous inertial neural networks (DINNs) with parameter uncertainties and time delays is studied. The main aim is to investigate the new fixed-time stability (FTS). In order to achieve the targets, first, by introducing the generalized variable transformation and differential inclusions theory, two kinds of drive–response differential inclusion systems are established. Based on the definition of FTS and inequality technologies, by constructing the Lyapunov–Krasovskii functional (LKF), whose derivative is allowed to be indefinite, new delay-dependent criteria shown by some simple inequalities are derived for the purposing of achieving the FTS based on the designed discontinuous control strategies. Moreover, the new settling time (ST) is given. Compared to the previous stability results on INNs, the results established and the approaches applied are absolutely new. Finally, examples are given to show the effectiveness of the established results.

This paper presents a class of Cohen–Grossberg neural networks (CGNNs) with discontinuous activations and time-varying delays. Firstly, under the framework of Filippov solution, we derive some general sufficient conditions to guarantee the global existence of the solutions to the proposed CGNNs with discontinuous activations and time-varying delays. Then, by constructing the new Lyapunov–Krasovskii functional, some new sufficient criteria are given to ascertain the globally exponential stability of the anti-periodic solution for the considered CGNNs with discontinuous activations and time-varying delays. To the authors’ knowledge, the results established in the paper are the only available results on the anti-periodic for the discontinuous CGNNs; some previously known results are extended and complemented. Finally, simulation results of two typical numerical examples are also delineated to demonstrate the effectiveness of our theoretical results.

This article aims to analyze the fixed-time synchronization of a class of discontinuous fuzzy inertial neural networks with time-varying delays based on the new improved fixed-time stability lemmas. First of all, by using the generalized variable transformation and Filippov solution theory, the discontinuities of the considered neural system can be coped with, and the error system is established. By relaxing the conditions of the$C$-regular Lyapunov function, two new fixed-time stability lemmas are proved via simple inequality techniques. The setting times are also estimated and are more accurate in comparison with the previous ones. Finally, one typical numerical example is carried out to verify the correctness and the advantages of the main results.

Abstract This article presents a class of neutral‐type Cohen‐Grossberg neural networks with discontinuous activations and mixed time delays. Based on the functional differential inclusions theory, inequality technique, and nonsmooth analysis theory with Lyapunovlike approach, some new sufficient criteria are given to ascertain the existence, uniqueness, and globally exponential stability of the antiperiodic solution. Finally, two topical numerical examples and the corresponding computer simulations are delineated to substantiate the correctness of our theoretical predictions. The obtained results of this article are new and complement some related earlier works.