Double Integral Inequality Research Articles

Generalization of the Fuzzy Fejér–Hadamard Inequalities for Non-Convex Functions over a Rectangle Plane

Integral inequalities with generalized convexity play a vital role in both theoretical and applied mathematics. The theory of integral inequalities is one of the branches of mathematics that is now developing at the quickest rate due to its wide range of applications. We define a new Hermite–Hadamard inequality for the novel class of coordinated ƛ-pre-invex fuzzy number-valued mappings (C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s) and examine the idea of C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s in this paper. Furthermore, using C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s, we construct several new integral inequalities for fuzzy double Riemann integrals. Several well-known results, as well as recently discovered results, are included in these findings as special circumstances. We think that the findings in this work are new and will help to stimulate more research in this area in the future. Additionally, unique choices lead to new outcomes.

Analyzing the Stochastic Stability of Neural Networks with Continuous-Time Semi-Markov Jump Parameters

This paper addresses the issue of stochastic stability for continuous-time semi-Markovian jump neural networks. Initially, according to the characteristics of the time-delay interval, the Lyapunov-Krasovskii functional (LKF) with the semi-Markov process is constructed to ensure the stability of the switching networks.his paper addresses the issue of stochastic stability for continuous-time semi-Markovian jump neural networks. Initially, according to the characteristics of the time-delay interval, the Lyapunov-Krasovskii functional (LKF) with the semi-Markov process is constructed to ensure the stability of the switching networks.T Next, the promoted double-integral inequality lemma is utilized to estimate the weak infinitesimal operator of the designed LKF, and this paper establishes a time-delay correlation criterion. Moreover, by the criterion is combined with the Lyapunov stability theory to make the system reach stability in the mean-square sense. Finally, the paper provides an example to demonstrate the effectiveness of the proposed approach.

A novel nonzero functional method to extended dissipativity analysis for neural networks with Markovian jumps

<abstract><p>This paper explored the topic of extended dissipativity analysis for Markovian jump neural networks (MJNNs) that were influenced by time-varying delays. A distinctive Lyapunov functional, distinguished by a non-zero delay-product types, was presented. This was achieved by combining a Wirtinger-based double integral inequality with a flexible matrix set. This novel methodology addressed the limitations of the slack matrices found in earlier research. As a result, a fresh condition for extended dissipativity in MJNNs was formulated, utilizing an exponential type reciprocally convex inequality in conjunction with the newly introduced nonzero delay-product types. A numerical example was included to demonstrate the effectiveness of the proposed methodology.</p></abstract>

Composite resilient state tracking and disturbance rejection control for periodic piecewise polynomial delay systems with multiform disturbances

In this study, we scrutinize the problems of resilient state tracking control (RSTC) and disturbance rejection (DR) for periodic piecewise polynomial delay systems in the presence of multiform disturbances and gain fluctuations. Specifically, we address both exosystem-induced disturbances and norm-bounded disturbances. Firstly, to tackle effectively the disturbances that are produced by exosystems, the disturbance observers are tailored, which estimates the disturbance in a precise manner. Secondly, the repercussions of norm-bounded disturbances can be lessened by accounting an extended dissipative performance. Moreover, by amalgamating the outcomes of the disturbance observer with the feedback-based tracking control, a DR-based RSTC is constructed. In addition, some parameter variations in the controller gain are taken into consideration in order to guarantee resiliency against unexpected flaws. Besides, the adequate conditions are derived in the form of linear matrix inequalities by making use of Lyapunov stability theory, new double integral inequality and extended Wirtinger’s inequality. Furthermore, the relation for acquiring the controller and observer gain matrices is formulated based on the established criteria. Finally, a numerical example with simulation depiction is presented to show the favour performance and capability of the designed controller.

Improved Integral Inequality Based on Free Matrices and Its Application to Stability Analysis of Delayed Neural Networks via Matrix-valued Cubic Polynomial Inequality

This paper deals with the stability analysis of delayed neural networks (DNNs). Firstly, an improved integral inequality based on free matrices (I3BFM) is newly proposed. The I3BFM gives the upper bound of the integral quadratic form including the state-related vector stacked by the state, its derivative, and its integration by introducing some free matrices. The upper bound of the I3BFM fully reflects the information not only on each term in the state-related vector but also on the cross-terms between the three terms. Secondly, the double integral Lyapunov-Krasovskii functional (LKF) of the integral quadratic form including the state-related vector is established to utilize the proposed I3BFM in the stability analysis of DNN. From the derivative of the double integral LKF, a matrix-valued cubic polynomial on a time-varying delay is generated in the process of stability analysis. Therefore, a matrix-valued cubic polynomial inequality is applied to make the matrix-valued cubic polynomial numerically tractable. Finally, the relaxed stability criteria utilizing the I3BFM and the double integral inequality are derived, and the superiority of the derived stability criteria is demonstrated by two well-known numerical examples.

Application of Double Integral Inequality for the Stability Analysis of the Singular Time-Varying Delay System

This paper aims to establish and analyze the stability of continuous-time linear descriptor systems with a time-varying delay, based on the famous method Lyapunov KRASOVSKI function and the Linear Matrix Inequality (LMI) technique. The founded theorem, not only allows the stability, but also the admissibility, so that the regularity and the impulse free of the studied descriptor system. The established criterion has been met using a neutral approach, which is the system transformation that checks singular time delayed system stability via a less complicated computation. Additionally to this new form, a major integral inequality lemma has also been used, due to its conservative results. The novelty of this idea is represented in the exploitation of the new double integral inequality proposed, in the admissibility analysis of singular time-delayed system via a neutral approach, which has been used to deliver an alternative delay dependent criterion in term of (LMI). The purpose of this idea is to check this admissibility of the said system through linear matrix inequality founded by programming it on the software numeric computing platform MATLAB. The originality of this article has been obtained by the combination of the double inequality technique and the neutral approach. At the end of this article, a numerical example has been provided using the LMI toolbox, in order to prove the validity and the efficiency of the proposed method. A comparison table will be presented to show the developed results in some other works compared with the proposed theorem, noting that the simulations of this example have been also performed in the MATLAB environment.

A Stimulus Generalization of Double Integral Inequalities by the way of Fractional Integrals

This study presents some of the latest results annexed to Hermite Hadamard inequality by utilizing double integrals by the way of Riemann-Liouville fractional integrals. Another aim of this article is to generalize some of the recent developments on Hermite Hadamard's type inequalities.

Passivity Analysis of Delayed Neural Networks Via Novel Integral Inequality

This paper studies the passivity analysis of delayed neural networks (NNs) via novel integral inequality. First of all, an improved double integral inequality is proposed by virtue of the zero-qualities and some other analytical techniques Secondly, more information about time-delayed and neuron activation function vector are considered in constructing the augmented Lyapunov-Krasovskii functional (LKF). Leant on the several methods mentioned above, some less conservative conditions are obtained. In the end, some numerical examples are provided to show the effectiveness and superiority over the derived results.

GENERALIZATION OF YANG–HARDY–HILBERT’S INTEGRAL INEQUALITY ON THE FRACTAL SET ℝ+α

Based on local fractional calculus theory, the Hölder double local fractional integral inequality with Weighted is proved. By using the methods of weight function and some analysis techniques on the fractal real set, a local fractional integral inequality with the best constant is given, which is generalization of Yang–Hardy–Hilbert’s inequality on the fractal set [Formula: see text] of fractal dimension [Formula: see text] and its equivalent form is considered.

Stochastic stability and extended dissipativity analysis for delayed neural networks with markovian jump via novel integral inequality

This paper studies the stochastic stability and extended dissipativity analysis for delayed Markovian jump neural networks (MJNNs) with partly unknown transition rates (PUTRs) using novel integral inequality. A new double integral inequality with augmented vector is introduced through inequality technique and the zero-valued equality approach, which can more efficiently estimate the derivative of the triple integral inequality. Next, an augmented Lyapunov-Krasovskii functional (LKF) with delay-product-type (DPT) is constructed. Besides, with the introduced integral inequality, the augmented LKF and some other analytical techniques, some less conservative extended dissipation conditions are obtained in the form of linear matrix inequality (LMI). Finally, several examples are provided to illustrate the effectiveness of the obtained results.

Weighted Cebysev Type Inequalities for Double Integrals and Application

The purpose of this article is to generalize Cebysev type inequalities for double integrals involving a weight function.By using an integral transform that is a weighted Montgomery identity, we obtained a generalized form of weighted Cebysev type inequalities in $L_m,, mgeq 1$ norm of differentiable functions. Also, we give some applications of the probability density function.

Strict dissipativity synchronization for delayed static neural networks: An event-triggered scheme

This article addresses the investigation of strict dissipativity synchronization for a class of static neural networks under an event-triggered scheme. An event-triggered scheme is recommended, it can upgrade the exhibition of system dynamics and diminishes the network communication burden at the same time. Firstly, an appropriate Lyapunov-Krasovskii functional (LKF) with double and triple integral terms with the details on both lower and upper bounds of the delay is completely designed. Secondly, under the single and double Auxillary function-based integral inequalities (SAFBII and DAFBII, respectively) and generalized free weight matrix approach, a new class of delay-dependent adequate condition is proposed, so that the error system is (Q,S,R)−γ− strict dissipative. A resilient distributed event-triggered control scheme is developed by this criterion in terms of linear matrix inequalities (LMIs). At last, simulation examples are provided to demonstrate the performance of the derived results.

New delay-dependent conditions for finite-time extended dissipativity based non-fragile feedback control for neural networks with mixed interval time-varying delays

This paper studies the delay-dependent conditions for finite-time extended dissipativity based non-fragile feedback control for neural networks with mixed interval time-varying delays. By applying Jensen’s inequality, an extended Jensen’s double integral inequality, and a free matrix form inequality to the Lyapunov–Krasovskii functional (LKF), delay-dependent conditions are derived and solved by the Matlab control toolbox in terms of linear matrix inequalities (LMIs). By stability criteria, this paper is less conservative than the other works. In addition, we demonstrate the advantage of our obtained methods by five numerical examples. One practical example shows a real-world approach: the quadruple-tank process system (QTPS).

Improved robust H ∞ stability analysis and stabilisation of uncertain and disturbed networked control systems with network-induced delay and packets dropout

This paper is concerned with stability analysis and stabilisation of uncertain Networked Control Systems (NCS) with network-induced delay and packet dropouts. This proposal aims at maximising the allowable upper bound of network-induced delay, together with the number of consecutive packet dropouts, from which uncertain and disturbed NCS plants are stable. A novel augmented Lyapunov–Krasovskii functionals (LKF), dependent on both the lower and upper bounds of the network-induced time-varying delay, is considered to derive the proposed delay-dependent LMI-based stability conditions. Relaxation of these conditions are obtained by exploiting simple and double integral Wirtinger's inequalities to introduce more degrees of freedom from the LKF terms than existing related conditions in the literature. Finally, numerical examples and simulation results are presented to illustrate the effectiveness and the conservatism improvement raised by this proposal compared to previous related results.

Improved Results on Delay dependent Stability Criteria of Neural Networks with Interval Time Varying Delay

This paper examines the stability issue of continuous Neural Networks with a time varying delay. A Lyapunov Krasovskii functional consisting of some simple augmented terms and delay dependent terms is constructed. While calculating the derivative of Lyapunov functional, various integral inequalities such as Auxiliary Function Based Integral Inequality, Wirtinger-based integral inequality and an extended Jensen double integral inequality are jointly adopted and hence in terms of linear matrix inequality a new delay dependent stability criterion is obtained. Two numerical examples are taken to show that the derived result is less conservative than some existing ones.

Enhanced Results on Stability Criteria for Linear Time Delay Systems with Distributed Delay via Relaxed Double Integral Inequality

This paper investigates the matter of stability criteria for linear time delay systems with distributed delay. Firstly, a relaxed double integral inequality is established to estimate the double integral terms appearing within the derivative of Lyapunov-Krasovskii functionals (LKFs) with a triple integral term. Unlike the recently introduced Jensen's inequalities, Wirtinger based integral inequalities, refined Jensen's inequalities and therefore the auxiliary function based integral inequalities the proposed relaxed integral inequality provides large feasible solution region and fewer conservative results. Secondly, by constructing an augmented Lyapunov-Krasovskii functional with a triple integral term, the robust stability criteria for linear time delay systems with distributed delay are given in terms of linear matrix inequalities (LMIs), which may be easily computed by the LMI toolbox of MATLAB. Finally, two numerical examples are performed to indicate the effectiveness of the proposed criterion.

Stabilization of Delayed Fuzzy Neutral-type Systems Under Intermittent Control

This study is concerned about the stabilization for delayed fuzzy neutral-type system (DFNTS) with uncertain parameters under intermittent control. Firstly, by constructing the augmented Lyapunov-Krasovskii functional (LKF) about different time delays along with single and double auxillary function-based integral inequalities (SAFBII, and DAFBII, respectively), a new class of delay-dependent adequate conditions are proposed, so that the robust fuzzy neutral-type system under consideration is guaranteed to be globally asymptotically stable (GAS). Secondly, the intermittent control (IC) is introduced to stabilize the system with mixed time-varying delays. In the view of inferred adequate conditions, the IC parameters are determined as for the arrangement of linear matrix inequalities (LMIs). It is noted that the strategies exploited in this work are apart from the other methods engaged in the literature, and the proposed conditions are less conservative. Finally, numerical examples are given to demonstrate the effectiveness of the developed techniques in this work. One of the practical applications is single-link robot arm (SLRA) model to show the viability and benefits of the structured intermittent control.

Extended Integral Inequality for Stability Analysis of Time-delay Systems

In this paper, we propose an extension of the generalized free-weighting-matrix based integral inequality, which is an enriched special case of the conventional inequality. This improved inequality requires less conservative criteria, since the vectors can be chosen independently and freely. By applying the proposed inequality combined with a double integral inequality to evaluate the derivative of Lyapunov-Krasovskii functional, a new stability criterion is formulated for a system with a time delay. To illustrate the effectiveness of the new stability criteria, we take three commonly used numerical examples. Simulation results confirm the effectiveness of the proposed approach.

Input–Output Finite-Time Reliable Static Output Control of Time-Varying System With Input Delay

This paper investigates input-output finite-time reliable static output feedback (SOF) control of a time-varying system under the influence of both input time delay and actuator failures. An actuator fault model consisting of linear and nonlinear faults is considered during the time-varying control process. The objective is to design a reliable SOF controller that can ensure input-output finite-time stability (IO-FTS) of the resulting closed-loop system. An augmented time-varying Lyapunov functional is constructed, in which some Lyapunov matrices are variable function of time t. By dividing the time interval and delay interval into equal segments, the matrix-valued functions are expressed by a linear interpolation formula. Moreover, combining with the single and double Wirtinger-based integral inequalities, delay-dependent IO-FTS conditions are derived. It is shown that the SOF control issue is solved in forms of linear matrix inequalities. In the end, the effectiveness is demonstrated by simulations.

Stability analysis of singular time-delay systems using the auxiliary function-based double integral inequalities

Recently, there have been a few developments reported on using the Wirtinger/free-matrix-based single integral inequality to stability problem of singular time-delay systems but there has been no report on the double ones. This paper presents an extension on applying the auxiliary function-based double integral inequality to the problem. Furthermore, an extension of the delay-dependent matrix technique into the single integral term of the Lyapunov–Krasovskii function to reduce more the conservatism has also been presented. By proposing an extended Lyapunov–Krasovskii functional (LKF) with triple integral terms and three delay-dependent matrices, a new delay-derivative-dependent stability criterion is derived. The effectiveness of the obtained result is illustrated through a numerical example.