Jensen Integral Inequality Research Articles

Does the application of Jensen's integral inequality and LMIs confirm exponential stability in delayed systems with gapped gamma distribution through augmented Lyapunov function?

This study is dedicated to a comprehensive exploration aimed at advancing our understanding of stability within dynamic systems. The focus is particularly on the intricate domain of delayed systems characterized by gapped gamma distributions. The primary objective of this investigation revolves around evaluating the pragmatic application and efficacy of Jensen's integral inequality in combination with the powerful analytical tools provided by Linear Matrix Inequalities (LMIs). This evaluation is crucial for rigorously assessing exponential stability within these complex systems. Central to our investigative framework is the strategic deployment of augmented Lyapunov functions. These functions play a crucial role in unraveling the intricate stability properties of delayed systems featuring gapped gamma distributions, allowing for a nuanced examination of their inherent stability characteristics under various conditions. The mathematical formulation crafted in this exploration intricately captures the interplay between the distinctive attributes of the gapped gamma distribution and the complex dynamics of the loop traffic flow model within the overarching delayed system. This interconnection serves as the fundamental basis for the stability analysis, providing insights into the interdependence of these key elements. The noteworthy contribution of this study lies in the systematic construction of a robust analytical framework explicitly tailored for stability assessment. A comprehensive investigation is undertaken to elucidate critical aspects, including the convergence rate and the attainment of asymptotic stability within the considered delayed system. Additionally, a dedicated simulation section, focusing on Vehicle Active Suspension Control, has been incorporated to further validate and showcase the applicability of the proposed methodology.

On the generalization of Hermite-Hadamard type inequalities for [formula omitted]-convex function via fractional integrals

The main motivation in this article is to prove new integral identities and related results. In this paper, we deal with E`-convex function, Hermite-Hadamard type inequalities, and Katugampola fractional integrals. Firstly, we established the Hermite-Hadamard type inequalities for E`-convex function via Katugampola fractional integrals operators. After that we obtain several new identities and give new results of Holder's inequality, Jensen's integral inequality, Young's inequality, and Power mean inequality for E`-convex function by using Katugampola fractional integral operators. We make connections of our results with various results recognized in the literature.

Quantized Proportional-Integral Tracking Control for Two-Stage Chemical Reactor Systems

Abstract This paper is concerned with the problem of proportional-integral tracking control of a two-stage chemical reactor system subject to time delays, disturbances, uncertainties, and input quantization. In this work, an improved equivalent-input-disturbance estimator is incorporated into the proportional-integral tracking control system to compensate for the disturbances in the addressed model. Moreover, to minimize the communication congestion in the control networks, the quantized control input signals are considered while designing the controller. Further, a robust stability condition for the addressed system is established in the form of linear matrix inequalities by employing asymmetric Lyapunov–Krasovskii functional and Jensen's integral inequalities. Moreover, in accordance with the derived conditions, the control and observer gain matrices are determined. Finally, a numerical example is provided to demonstrate the validity of the proposed control scheme.

Robust stability of uncertain stochastic switched inertial neural networks with time‐varying delay using state‐dependent switching law

In this paper, the problem of robust stability analysis for a type of uncertain stochastic switched inertial neural networks (SSINNs) with time‐varying delay is investigated. First, the original second‐order system is converted into first‐order differential equations using the variable transformation method. Next, some sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained using Lyapunov‐Krasovskii functional (LKF), state‐dependent switching (SDS) method, and Jensen's integral inequality for estimating integral inequalities so that the augmented system is robust, global, and asymptotically stable in the mean square for the uncertain SSINNs with time‐varying delay. It is shown that the stability of the above considered system composed of all unstable subsystems can be achieved by using the SDS law. Finally, two numerical simulations are provided to demonstrate the effectiveness of the proposed SDS law.

New delay-range-dependent exponential stability criterion and $ H_\infty $ performance for neutral-type nonlinear system with mixed time-varying delays

<abstract><p>For a neutral system with mixed discrete, neutral and distributed interval time-varying delays and nonlinear uncertainties, the problem of exponential stability is investigated in this paper based on the $ H_\infty $ performance condition. The uncertainties are nonlinear time-varying parameter perturbations. By introducing a decomposition matrix technique, using Jensen's integral inequality, Peng-Park's integral inequality, Leibniz-Newton formula and Wirtinger-based integral inequality, utilization of a zero equation and the appropriate Lyapunov-Krasovskii functional, new delay-range-dependent sufficient conditions for the $ H_\infty $ performance with exponential stability of the system are presented in terms of linear matrix inequalities. Moreover, we present numerical examples that demonstrate exponential stability of the neutral system with mixed time-varying delays, and nonlinear uncertainties to show the advantages of our method.</p></abstract>

Some generalizations of reverse Hardy-type inequalities via Jensen integral inequality on time scales

Some generalizations of reverse Hardy-type inequalities via Jensen integral inequality on time scales

Refinements of Ostrowski Type Integral Inequalities Involving Atangana–Baleanu Fractional Integral Operator

In this article, first, we deduce an equality involving the Atangana–Baleanu (AB)-fractional integral operator. Next, employing this equality, we present some novel generalization of Ostrowski type inequality using the Hölder inequality, the power-mean inequality, Young’s inequality, and the Jensen integral inequality for the convexity of |Υ|. We also deduced some new special cases from the main results. There exists a solid connection between fractional operators and convexity because of their fascinating properties in the mathematical sciences. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. It is assumed that the results presented in this article will show new directions in the field of fractional calculus.

Normalized fractional inequalities for continuous random variables

We introduce new normalized fractional integral concepts on random variables. We also introduce the fractional coupled random variables. The Jensen integral inequality is generalized. Also, some fractional bounds estimating the expectation and variance are delivered. At the end, some other results related to the coupled random variables are proved. For this work, some results of the papers [On some statistical and probability inequalities. Journal of Inequalities and Special Functions, 2016] and [Some inequalities for the expectation and variance of a random variable whose PDFs are absolutely continuous using a pre-Chebychev inequality. Tamkang Journal of Mathematics, 2001] are deduced as some special cases.

Reverse Jensen Integral Inequalities for Operator Convex Functions in Terms of Fréchet Derivative

Reverse Jensen Integral Inequalities for Operator Convex Functions in Terms of Fréchet Derivative

Reverses Hardy-type inequalities via Jensen integral inequality

The integral inequalities concerning the inverse Hardy inequalities have been studied by a large number of authors during this century, of these articles have appeared, the work of Sulaiman in 2012, followed by Banyat Sroysang who gave an extension to these inequalities in 2013. In 2020 B. Benaissa presented a generalization of inverse Hardy inequalities. In this article, we establish a new generalization of these inequalities by introducing a weight function and a second parameter. The results will be proved using the Hölder inequality and the Jensen integral inequality. Several the reverses weighted Hardy’s type inequalities and the reverses Hardy’s type inequalities were derived from the main results.

Finite-Time MixedH∞/Passivity for Neural Networks With Mixed Interval Time-Varying Delays Using the Multiple Integral Lyapunov-Krasovskii Functional

In this article, we consider the finite-time mixed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> /passivity, finite-time stability, and finite-time boundedness for generalized neural networks with interval distributed and discrete time-varying delays. It is noted that this is the first time for studying in the combination of H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> , passivity, and finite-time boundedness. To obtain several sufficient criteria achieved in the form of linear matrix inequalities (LMIs), we introduce an appropriate Lyapunov-Krasovskii function (LKF) including single, double, triple, and quadruple integral terms, and estimating the bound of time derivative in LKF with the use of Jensen's integral inequality, an extended single and double Wirtinger's integral inequality, and a new triple integral inequality. These LMIs can be solved by using MATLAB's LMI toolbox. Finally, five numerical simulations are shown to illustrate the effectiveness of the obtained results. The received criteria and published literature are compared.

Stability analysis for delayed neural networks based on a generalized free-weighting matrix integral inequality

This paper investigates the stability problem of neural networks (NNs) with time-varying delay. Firstly, a new augmented vector and suitable Lyapunov–Krasovskii Functional (LKF) considering activation function are constructed by using more information of time delay. Secondly, a generalized free-weighting matrix integral inequality (GFMII) is chosen to estimate the derivative of single integral terms more accurately. Meanwhile, Jensen integral inequality and improved convex combination are combined to estimate integral terms with activation function; as a result, a novel stability criterion with less conservatism is established. Finally, two numerical examples are employed to illustrate the effectiveness of proposed methods.

Integral inequalities via Raina\u2019s fractional integrals operator with respect to a monotone function

We establish certain new fractional integral inequalities involving the Raina function for monotonicity of functions that are used with some traditional and forthright inequalities. Taking into consideration the generalized fractional integral with respect to a monotone function, we derive the Grüss and certain other associated variants by using well-known integral inequalities such as Young, Lah–Ribarič, and Jensen integral inequalities. In the concluding section, we present several special cases of fractional integral inequalities involving generalized Riemann–Liouville, k-fractional, Hadamard fractional, Katugampola fractional, (k,s)-fractional, and Riemann–Liouville-type fractional integral operators. Moreover, we also propose their pertinence with other related known outcomes.

Ohlin-Type Theorem for Convex Set-Valued Maps

A counterpart of the Ohlin theorem for convex set-valued maps is proved. An application of this result to obtain some inclusions related to convex set-valued maps in an alternative unified way is presented. In particular counterparts of the Jensen integral and discrete inequalities, the converse Jensen inequality and the Hermite–Hadamard inequalities are obtained.

Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances

<p style='text-indent:20px;'>This paper investigates the delay-distribution-dependent exponential synchronization problem for a class of chaotic neural networks with mixed random time-varying delays as well as restricted disturbances. Given the probability distribution of the time-varying delay, stochastic variable that satisfying Bernoulli distribution is formulated to produce a new system which includes the information of the probability distribution. Based on the Lyapunov-Krasovskii functional method, the Jensen's integral inequality theory and linear matrix inequality (LMI) technique, several delay-distribution-dependent sufficient conditions are developed to guarantee that the chaotic neural networks with mixed random time-varying delays are exponentially synchronized in mean square. Furthermore, the derived results are given in terms of simplified LMI, which can be straightforwardly solved by Matlab. Finally, two numerical examples are proposed to demonstrate the feasibility and the effectiveness of the presented synchronization scheme.

Enhanced stability results for generalized neural networks with time-varying delay

This paper deals with the stability analysis problem for generalized neural network with time-varying delay. A proper Lyapunov-Krasovskii functional (LKF) with novel delay-product-dependent term is constructed, which fully considers the features of the improved Jensen integral inequality. Based on the new proposed LKF and a more general integral inequality issued from the improved Jensen one, an enhanced delay-dependent stability criterion is developed in terms of linear matrix inequality (LMI). Furthermore, a more refined stability criterion is established by relaxing the positive definiteness of the LKF. The effectiveness of the theoretical results are demonstrated by four numerical examples.

Improved Results on Stability Analysis for Delayed Neural Network

This paper deals with the delay-dependent stability analysis problem for neural network with a time-varying delay. A proper Lyapunov-Krasovskii functional (LKF) is established by revealing the features of the improved Jensen integral inequality and considering two complementary integral couples with more cross information. Based on the improved Jensen inequality, a generalized integral inequality involving more free matrices is developed. With the help of the new LKF and integral inequality, some improved stability conditions with less conservatism are derived in terms of linear matrix inequality (LMI). The efficiency of theoretical results is verified by three typical numerical examples.

Converses of the Jensen inequality derived from the Green functions with applications in information theory

Jensen integral inequality has got much importance regarding their applications in different fields of mathematics. In this paper, two converses of Jensen integral inequality for convex function are obtained. The results are applied to establish converses of Hölder and Hermite‐Hadamard inequalities as well. At the end, some useful applications in information theory of the obtained results are given. The idea used in this paper may inculcate further research.

Finite‐time reliable dissipative control of neutral‐type switched artificial neural networks with non‐linear fault inputs and randomly occurring uncertainties

AbstractThis work reports the result of finite‐time reliable dissipative control for switched neutral‐type based neural networks subject to input nonlinearities, time‐delay and randomly occurring perturbations. In particular, a novel control model consist of both linear and non‐linear fault inputs is designed for reliable control. In addition, randomly occurring uncertainty caused by stochastic variables satisfying the Bernoulli distribution is considered. Based on Lyapunov stability theory, Jensen's integral inequality technique and average dwell time method, sufficient criteria for finite‐time boundedness of resulting neural networks are obtained. Furthermore, the obtained boundedness theory are enhanced to solve the finite‐time dissipative problem of the considered systems through reliable control against mixed actuator failures. The obtained sufficient conditions are constructed in the form of linear matrix inequalities, which can be facilitated by using some standard numerical Matlab packages. At last, numerical simulations are presented to demonstrate the effectiveness of the considered theory.

A delay-dependent anti-windup compensator for wide-area power systems with time-varying delays and actuator saturation

In this paper, a delay-dependent anti-windup compensator is designed for wide-area power systems to enhance the damping of inter-area low-frequency oscillations in the presence of time-varying delays and actuator saturation using an indirect approach. In this approach, first, a conventional wide-area damping controller is designed by using H&#x221E output feedback with regional pole placement approach without considering time-varying delays and actuator saturation. Then to mitigate the effect of both time-varying delays and actuator saturation, an add-on delay-dependent anti-windup compensator is designed. Based on generalized sector conditions, less conservative delay-dependent sufficient conditions are derived in the form of a linear matrix inequality &#x0028 LMI &#x0029 to guarantee the asymptotic stability of the closed-loop system in the presence of time-varying delays and actuator saturation by using Lyapunov-Krasovskii functional and Jensen integral inequality. Based on sufficient conditions, the LMI-based optimization problem is formulated and solved to obtain the compensator gain which maximizes the estimation of the region of attraction and minimizes the upper bound of L2-gain. Nonlinear simulations are performed first using MATLAB &#x002F Simulink on a two-area four-machine power system to evaluate the performance of the proposed controller for two operating conditions, e.g., 3-phase to ground fault and generator 1 terminal voltage variation. Then the proposed controller is implemented in real-time on an OPAL-RT digital simulator. From the results obtained it is verified that the proposed controller provides sufficient damping to the inter-area oscillations in the presence of time-varying delays and actuator saturation and maximizes the estimation of the region of attraction.