Jensen's Inequality Research Articles

Resilient Distributed Voltage Synchronization of CI Networks Under Denial of Service Attacks

This brief proposes a resilient distributed sampled-data control method to achieve the voltage synchronization of connected inverter (CI) networks against denial of service (DoS) attacks. In practical CI networks, the continuous communication among cooperative and connected inverters may be result in high communication traffic, which is also vulnerable to the DoS attacks. To this end, the proposed distributed resilient sampled-data control method can synchronize CIs' output voltages to track the reference state exponentially, even if the sparse communication is subject to DoS attacks. Specifically, in order to effectively deal with the time-varying sampling and against malicious attacks simultaneously, the sampled-data CI system is transformed into the switched time-delay CI system. By employing the Lyapunov stability theory, and the Jensen's Inequality method, the criteria for sampling interval boundedness considering DoS attacks is derived for the stable operation of closed-loop CI system. Finally, simulation results are carried out in MATLAB/SimPower Systems to verify the effectiveness of the proposed control method.

Statistical Learning Based Joint Antenna Selection and User Scheduling for Single-Cell Massive MIMO Systems

Large number of antennas and radio frequency (RF) chains at the base stations (BSs) lead to high energy consumption in massive MIMO systems. Thus, how to improve the energy efficiency (EE) with a computationally efficient approach is a significant challenge in the design of massive MIMO systems. With this motivation, a learning-based stochastic gradient descent algorithm is proposed in this article to obtain the optimal joint uplink and downlink EE with joint antenna selection and user scheduling in single-cell massive MIMO systems. Using Jensen's inequality and the characteristics of wireless channels, a lower bound on the system throughput is obtained. Subsequently, incorporating the power consumption model, the corresponding lower bound on the EE of the system is identified. Finally, learning-based stochastic gradient descent method is used to solve the joint antenna selection and user scheduling problem, which is a combinatorial optimization problem. Rare event simulation is embedded in the learning-based stochastic gradient descent method to generate samples with very small probabilities. In the analysis, both perfect and imperfect channel side information (CSI) at the BS are considered. Minimum mean-square error (MMSE) channel estimation is employed in the study of the imperfect CSI case. In addition, the effect of a constraint on the number of available RF chains in massive MIMO system is investigated considering both perfect and imperfect CSI at the BS.

Heat and Laplace type equations with complex spatial variables in weighted Fock spaces

In a recent book co-authored by the authors of this article, we studied by semigroup theory methods several classical evolution equations, including the heat and Laplace equations, with real time variable and complex spatial variable, under the hypothesis that the boundary function belongs to the space of analytic functions in the open unit disk and continuous in the closed unit disk, endowed with the uniform norm. Also, in a subsequent paper, the authors have extended the results for the heat and Laplace equations in weighted Bergman spaces on the unit disk. The purpose of this article is to show that the semigroup theory methods work for these two evolution equations of complex spatial variables, under the hypothesis that the boundary function belongs to the weighted Fock space on \(\mathbb{C}\), \(F_{\alpha }^p(\mathbb{C})\), with \(1\leq p<+\infty \), endowed with The \(L^p\)-norm. Also, the case of several complex variables is considered. The proofs use the Jensen's inequality, Fubini's theorem for integrals and the \(L^p\)-integral modulus of continuity.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/109/abstr.html

Practical Stabilization of Networked Takagi-Sugeno Fuzzy Systems via Improved Jensen Inequalities.

This work addresses the problem of aperiodically sampled control for the networked Takagi-Sugeno (T-S) fuzzy systems, where the aperiodically sampled input is generated by a periodic sampler and an event-triggered mechanism (ETM). The purpose of ETM is used to reduce the computational and communication burdens. For guaranteeing controller robustness, the practical stability of T-S fuzzy systems is considered by using the Lyapunov method and linear matrix inequality (LMI) technique. As one of the most powerful inequalities for deriving stability criteria using LMIs, Jensen's inequality has recently been improved by various authors for the stability analysis of delayed systems. However, these results are conservative to obtain lower bounds for integrals with an exponential term. Inspired by this, improved integral inequalities are derived in this work, and they are applied to obtain practical stability criteria for aperiodically sampled control. Finally, a numerical example on flight control of a helicopter is given to illustrate the effectiveness of the obtained practical stability criteria. Furthermore, the effectiveness of the improved Jensen inequalities on the exponential stability criteria is illustrated by numerical comparisons.

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.

Inequalities of Hardy Type via Superquadratic Functions with General Kernels and Measures for Several Variables on Time Scales

We use the properties of superquadratic functions to produce various improvements and popularizations on time scales of the Hardy form inequalities and their converses. Also, we include various examples and interpretations of the disparities in the literature that exist. In particular, our findings can be seen as refinements of some recent results closely linked to the time-scale inequalities of the classical Hardy, Pólya-Knopp, and Hardy-Hilbert. Some continuous inequalities are derived from the main results as special cases. The essential results will be proved by making use of some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality on time scales.

Synchronization of Inertial Neural Networks With Time-Varying Delays via Quantized Sampled-Data Control.

This article addresses the quantized sampled-data (QSD) synchronization for inertial neural networks (INNs) with heterogeneous time-varying delays, in which the sampled-data control and state quantization effect have been considered. By utilizing a proper variable substitution to transform the original system into a first-order differential system, choosing a new Lyapunov-Krasovskii functional (LKF) containing both the continuous terms and the discontinuous terms, and applying Jensen inequality and an improved reciprocally convex inequality to estimate the derivative of the LKF, the sufficient conditions for QSD synchronization for INNs are newly obtained in terms of linear matrix inequalities (LMIs), and the desired QSD controllers are designed by solving a set of LMIs. Finally, three numerical examples are provided to validate the effectiveness and benefit of the proposed results.

Bipartite synchronization of coupled Lurie networks with signed graph and time-varying delay

This paper addresses the bipartite leader-following and leaderless synchronization in a signed network composed by an array of coupled Lurie systems with time-varying delay, where the signed network contains both the non-delay coupling term and the delay coupling term. By using the pinning control strategy and a gauge transformation to transform the problem of bipartite synchronization for signed network into that of synchronization of unsigned network, choosing an appropriate Lyapunov–Krasovskii functional (LKF), applying Jensen inequality and reciprocally convex inequality to estimate the derivative of the LKF, several sufficient conditions, which guarantee the bipartite leader-following and leaderless synchronization for the coupled Lurie networks with time-varying delay, are newly obtained in terms of linear matrix inequalities (LMIs). When the delay coupling term is not considered, some criteria in terms of low-dimensional LMIs are established by using M-matrix theory. A numerical example is provided to illustrate the effectiveness of the proposed results.

Constant and fluctuating temperature acclimations have similar effects on phenotypic plasticity in springtails.

Much interest exists in the extent to which constant versus fluctuating temperatures affect thermal performance traits and their phenotypic plasticity. Theory suggests that effects should vary with temperature, being especially pronounced at more extreme low (because of thermal respite) and high (because of Jensen's inequality) temperatures. Here we tested this idea by examining the effects of constant temperatures (10 to 30°C in 5°C increments) and fluctuating temperatures (means equal to the constant temperatures, but with fluctuations of ±5°C) temperatures on the adult (F2) phenotypic plasticity of three thermal performance traits - critical thermal minimum (CTmin), critical thermal maximum (CTmax), and upper lethal temperature (ULT50) in ten species of springtails (Collembola) from three families (Isotomidae 7 spp.; Entomobryidae 2 spp.; Onychiuridae 1 sp.). The lowest mean CTmin value recorded here was -3.56±1.0°C for Paristoma notabilis and the highest mean CTmax was 43.1±0.8°C for Hemisotoma thermophila. The Acclimation Response Ratio for CTmin was on average 0.12°C/°C (range: 0.04 to 0.21°C/°C), but was much lower for CTmax (mean: 0.017°C/°C, range: -0.015 to 0.047°C/°C) and lower also for ULT50 (mean: 0.05°C/°C, range: -0.007 to 0.14°C/°C). Fluctuating versus constant temperatures typically had little effect on adult phenotypic plasticity, with effect sizes either no different from zero, or inconsistent in the direction of difference. Previous work assessing adult phenotypic plasticity of these thermal performance traits across a range of constant temperatures can thus be applied to a broader range of circumstances in springtails.

On a version of fuzzy Jensen inequality based on the measure of level-set

In this paper, we present a sufficient condition for the continuity of distribution function α→μ(fα) based on a condition of the measure of level-sets. From this result, an estimate of Jensen-type inequality for Sugeno integral is stated. Finally, it is applied to establish Jensen-type estimates for several kinds of fuzzy numbers.

Stability analysis of neural network controller based on event triggering

In this paper, we investigated the stability of continuous-time nonlinear systems and designed a Neural-Network-Controller based event-triggered. First, we designed a system delay model for the transmission time-delay contained in the system. Based on the designed time-delay model, the input delay method was used to convert the synchronous controller solution problem into a stability problem for the corresponding time-delay system. Second, by using a piecewise Lyapunov-Krasovskii functional combined with the Jensen's inequality, both the stability condition of the continuous-time nonlinear systems and the solution method of sampling controller were given. The designed sampling controller is able to update the control parameters only when the triggering rules are fulfilled, thus reducing the transmission rate of the network. Finally, the effectiveness of the proposed method was verified by numerical simulation.

Connectivity of UAV Swarms in 3D Spherical Spaces Under (Un)Intentional Ground Interference

This paper analyzes the wireless connectivity of an Unmanned Aerial Vehicle (UAV) swarm in the presence of (un)intentional external interference from the ground. Different from existing studies, the swarm UAVs fly independently around a given three-dimensional (3D) location and are all within a 3D spherical space. Closed-form bounds are delivered for the average outage probability of a UAV from its nearest neighbor in the swarm, and the density of the swarm, which allows the swarm to operate uninterruptedly in the presence of the interference. Our analysis involves closed-form approximations of the instantaneous outage probability of a UAV from its nearest neighbor by using the first-order Marcum Q-function and the zero-th order modified Bessel function of the first kind. The analysis also involves applying Jensen's inequality to the instantaneous outage probability to bound the average outage probability and the density of the swarm. Corroborated by simulations, our analysis is accurate, and useful to evaluate the impact of external interference on the connectivity of UAV swarms. Interesting insights are shed on the connectivity and coverage of the UAV swarm.

The Jensen's inequality and functional form of Jensen's inequality for 3-convex functions at a point

The Jensen's inequality and functional form of Jensen's inequality for 3-convex functions at a point

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.

Some Properties and Inequalities for the h,s-Nonconvex Functions

The purpose of this paper is to introduce the notion of strongly h,s-nonconvex functions and to present some basic properties of this class of functions. We present Schur inequality, Jensen inequality, Hermite–Hadamard inequality, and weighted version of the Hermite–Hadamard inequality.

Decentralized stabilization of a class of large scale networked control systems based on modified event-triggered scheme

In this paper, a new method is proposed to design a decentralized $$H_\infty $$ dynamic output feedback controller for a class of large scale event-triggered networked control systems. The networked control system is modeled based on a modified event-triggered scheme. In addition to the time interval between triggers, the characteristics of network communication are also considered. The time delay and packet loss are two important phenomena and are inevitable in networked control systems. A time delay model is presented to collect all types of delay and packet loss in the event-triggered networked control system. Also, dynamic delays between subsystems are considered. Based on this model, some new sufficient conditions are given to guarantee the stability and the $$H_\infty $$ performance, based on LKF and integral estimation. The mixed Wirtinger–Jensen inequality is proposed to derive less conservative synthesis conditions in terms of linear matrix inequalities. A decentralized event-triggered dynamic output feedback controller is designed based on the required stability conditions. Simulation results of a two coupled inverted pendulum and continuous stirred-tank reactors show the effectiveness of the proposed method, which guarantees the asymptotical stability with less number of data packets and the more maximum allowable time interval between packet releases, in comparison with the common networked control schemes.

Jensen's inequalities for set-valued and fuzzy set-valued functions

Being an important part of classical analysis, Jensen's inequality has drawn much attention recently. Due to its generality, the inequality based on non-additive integrals appears in many forms, such as Sugeno integrals, Choquet integrals and pseudo-integrals. As a well-known generalization of classical one, the set-valued analysis is frequently applied to the research of mathematical economy, control theory and so on. Thus, it is of great necessity to generalize the set-valued case. Motivated by the pioneering work of Costa's Jensen's fuzzy-interval-valued inequality and Štrboja et al.'s Jensen's set-valued inequality based on Aumann integrals and pseudo-integrals respectively, this paper focuses particularly on proving certain kinds of Jensen's set-valued inequalities and fuzzy set-valued inequalities. These inequalities consist of two families: the related convex (or concave) function is a set-valued or fuzzy set-valued function and the integrand is a real-valued function; the related convex (or concave) function is a real-valued function and the integrand is a set-valued or fuzzy set-valued function. Particularly, Jensen's interval-valued and fuzzy-interval-valued inequalities, including Costa's, are obtained as corollaries.

A Jensen–Sherman type inequality with a control map for G-invariant convex functions

In this paper, the standard Jensen inequality for two points and convex functions is generalized to a version for four points satisfying Sherman’s majorization condition and for G-invariant convex functions, where G is a subgroup of the orthogonal group acting on an inner product space. A control map is used. A comparison of the obtained inequality and Jensen inequality is presented.

Finite-time exponential synchronization of reaction-diffusion delayed complex-dynamical networks

<p style='text-indent:20px;'>This investigation looks at the issue of finite time exponential synchronization of complex dynamical systems with reaaction diffusion term. This reort studies complex networks consisting of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> straightly and diffusively coupled networks. By building a new Lyapunov krasovskii functional (LKF), using Jensens inequality and convex algorithms approach stability conditions frameworks are determined. At last, a numerical precedent is given to demonstrate the practicality of the theoretical results.

A New Refinement of the Jensen Inequality with Applications in Information Theory

The Jensen inequality is one of the most important inequalities in theory of inequalities, and numerous results are devoted to this inequality. This inequality has many applications in several fields. This article is devoted to present a new interesting refinement of the well-known Jensen inequality and to give applications for means. The paper also aims to achieve numerous applications in information theory; in particular, a comprehensive detail has been given for Zipf’s law and obtained bounds for Zipf–Mandelbrot entropy. At the end of the article, a more general refinement of Jensen inequality is presented associated with m finite sequences.