Differential Inequalities Research Articles

This paper is devoted to finite-time output synchronisation for a class of complex networks with output couplings. A composite strategy combining impulsive state control with finite-time output control is applied to achieve finite-time output synchronisation. The non-invertible control input matrix is processed through matrix factorisation. With the help of a finite-time stability lemma and this factorisation, a finite-time output synchronisation criterion is established through a differential inequality with impulsive effect and Lyapunov stability. A numerical simulation is supplied to validate the efficacy of the developed control strategy.

In this work, we deal with the analysis of some qualitative properties of solutions to a class of fractional pseudo-parabolic equations involving nonlocal nonlinearity. We first establish the existence of local solutions to the problem. Then, we prove the existence and uniqueness of global solutions, as well as the long-time behavior of these global solutions. Next, we prove the blow-up phenomenon at finite time occurs when the initial datum lies in the unstable set, that is, I ( u 0 ) < 0 , without requiring the subcritical initial energy condition J ( u 0 ) ≤ d . Furthermore, we establish an upper bound for the blow-up time and derive both upper and lower bounds for the decay rate of the solutions. Finally, we show the existence of ground-state solutions to the corresponding stationary equation and investigate the relationship between global solutions and ground-state solutions. Our arguments are based on the Galerkin approximation method, the contraction mapping principle, variational methods combined with the Hardy–Littlewood–Sobolev inequality, and a modified differential inequality.

Existence and stability analysis of evolutionary differential hemivariational inequalities in Banach spaces

The purpose of this paper is to investigate an abstract system, a combination of a nonlinear evolutionary differential inclusion and a first-order evolutionary variational-hemivariational inequality (EDVHVI, for short) in the framework of Banach spaces. The central idea of our present paper consists of four parts. Initially, we are concerned with the results on the existence of solutions and the properties which involve the boundedness and continuous dependence results of the solution set to an evolutionary variational-hemivariational inequality (EVHVI). Subsequently, the second goal further studies the existence of mild solutions to EDVHVI by employing the fixed point theorem. Next, the existence of pullback attractors for the multivalued processes driven by EDVHVI is investigated in a general setting. At the end, we give a specific application about a comprehensive dynamic system generated by partial differential inclusions with obstacle effect, which is shown to substantiate our theory towards the conclusions.

In real-world applications, finite time convergence to a desired Lyapunov stable equilibrium is often necessary. This notion of stability is known as finite time stability and refers to systems in which the state trajectory reaches an equilibrium in finite time. This paper explores the notion of finite time stability in probability within the context of nonlinear stochastic dynamical systems. Specifically, we introduce sufficient conditions based on Lyapunov methods, utilizing Lyapunov functions that satisfy scalar differential inequalities involving fractional powers for guaranteeing finite time stability in probability. Then, we address the finite time optimal control problem by developing a framework for designing optimal feedback control laws that achieve finite time stochastic stability of the closed-loop system using a Lyapunov function that also serves as the solution to the steady-state stochastic Hamilton–Jacobi–Bellman equation.

Starlikeness for Bounded Analytic Functions and Reciprocal Starlikeness for a Class Involving a Differential Inequality

In this article, we address the error estimation problem of quasi-synchronization for a class of multilayer dynamical networks. The proposed network model simultaneously accounts for interlayer and intralayer time-varying coupling structures, network directionality, and interlayer communication delays. To achieve synchronization in a cost-effective manner, we design a novel pinning impulsive control strategy that leverages large-scale impulse delay information together with the number of pinned nodes. By employing an iterative algorithm, we establish a new delay-dependent impulsive differential inequality, which precisely characterizes the convergence domain and provides flexibility in the choice of impulse delays. Then, some quasi-synchronization criteria are derived to guarantee convergence of multilayer networks within a prescribed error level, and explicit analytical expressions for the synchronization error bounds are obtained. Finally, to demonstrate the practical applicability, the proposed criteria are applied to the synchronization of multilayer single-link robot arm networks under error bounds, with numerical examples validating the effectiveness of the method.

Robust synchronization of reaction-diffusion memristive neural networks with parameter uncertainties and general couplings.

This paper investigates a class of singularly perturbed boundary value problems with multiple turning points, exemplified by ε 2 y ″ − x ( x − k ) y ′ = x 2 , − 1 ≤ x ≤ 1 . By extending A.M.Il'in's matching method, we systematically construct higher-order asymptotic solutions for three critical cases (k = −1, 0, 1) and analyze zeroth-order approximations for − 1 < k < 0 and 0<k<1. The existence of solutions and error estimation are rigorously established via differential inequality theory. Numerical validations using Mathematica and MATLAB fully confirm the theoretical predictions.

This paper investigates the collaborative obstacle avoidance control of multiple autonomous underwater vehicles (AUVs) in underwater environments with communication delays and intermittent connectivity. Firstly, a novel time-delay piecewise differential inequality incorporating an exponential decay term is established, which systematically integrates state delay, intermittent control strategies, and event-triggered mechanisms. Secondly, the traditional request-response mechanism is replaced by a broadcast communication protocol, significantly reducing the requirement for continuous inter-AUV communication and enhancing overall communication efficiency. Thirdly, the obstacle avoidance problem of multiple AUVs is addressed through the implementation of a nominal-optimized controller. Obstacle avoidance safety conditions between unmanned vehicles and obstacles are derived by employing a zeroing control barrier function (CBF). Based on these safety conditions and input constraints, a quadratic programming (QP) framework is formulated to dynamically optimize control signals in real time, thereby ensuring the safe operation of the multi-AUV system. Finally, the efficacy of the proposed control method is validated through comprehensive simulation results, demonstrating its robustness and practical performance.

Energy access and utilization remain highly unequal across sub-Saharan Africa (SSA) countries, despite the region's vast natural resources and growing energy needs. This study examines the differential inequalities in energy access and utilization in selected SSA countries, focusing on demand and supply-side constraints, technological opportunities, and the role of government and private sector interventions. The review paper highlights the persistent energy poverty affecting over 600 million people in SSA, particularly in rural areas, where reliance on traditional biomass remains prevalent. Infrastructure deficiencies, high energy costs, and inadequate policy frameworks further exacerbate these inequalities. The paper underscores the critical role of governments in formulating effective energy policies, implementing subsidies, and fostering public-private partnerships to expand sustainable energy access. The private sector's involvement in financing and deploying decentralized energy solutions is identified as a key driver of progress. Recommendations include strengthening policy and regulatory frameworks, expanding regional power pools, investing in decentralized energy solutions, and promoting financial inclusion through innovative funding mechanisms. By addressing these challenges, SSA can move towards achieving equitable and sustainable energy access, fostering economic growth, and improving overall quality of life as key objectives of achieving the sustainable development goals (SDGs).

ABSTRACTIn this paper, a class of fourth‐order hyperbolic equation with logarithmic nonlinearity is considered. It is proved that the solutions blow up in finite time by developing the method of Levine concavity and using some differential inequality techniques. Meanwhile, the lifespan of the solutions is estimated. Furthermore, we establish an explicit energy decay rate.

A problem of designing an output feedback controller to improve transient performance in uncertain linear time-varying systems subjected to bounded external disturbances is considered. For systems with the dynamics specified by polytopic uncertainties, we design linear reduced-order dynamic controllers to bound the system state trajectories below a specified threshold over a predefined finite time interval. Using the notion of dissipativity, we also ensure that the system is robust with respect to bounded external disturbances over the finite time interval. Sufficient conditions for the uncertain closed-loop system trajectories to simultaneously satisfy finite-time boundedness and dissipativity to external disturbances are given in terms of parameter-dependent differential matrix inequalities. Further, parameter-independent differential linear matrix inequality (DLMI) conditions are presented to design time-varying control gains. The proposed dissipativity-based approach is less conservative as it generalises the procedure of designing finite-time robust controllers, encompassing various disturbance attenuation performance criteria like passivity and finite L 2 -gain. The applicability of the proposed approach is demonstrated through numerical simulation examples.

Fractional differential inequalities have emerged as powerful tools for modeling and analyzing dynamic systems with fractional-order derivatives, offering a sophisticated framework to capture the complexities of real-world processes. Among the various analytical techniques, the comparison principle stands out as a fundamental approach in understanding the behavior of solutions to fractional differential inequalities. This study focuses on the development and analysis of comparison principles for some of the fractional differential inequalities involving the variable-order Caputo fractional derivative which is a generalization of the classical Caputo derivative that allows the order of differentiation to vary with respect to time or space. Such flexibility is important for modeling systems whose memory characteristics change over time or space. We formulate both weak and strong versions of the comparison principle with variable order Caputo fractional derivative. Our approach combines analytical techniques from fractional calculus and the theory of differential inequalities to establish some results. To have the applicability and relevance of our theoretical work, we provide an example demonstrating the effectiveness of the proposed comparison theorems. The findings of this paper not only contribute to the theoretical advancement of fractional differential inequalities with variable order but also applicable to systems where dynamic memory effects are prominent.

This paper investigates a class of coupled parabolic equations featuring nonlocal terms subject to Robin boundary conditions. We introduce novel conditions on the initial data, leveraging differential inequality techniques in conjunction with auxiliary functions to derive both upper and lower bounds for the blow-up time. Building upon the established upper bound, we further refine our analysis by integrating the sub- and supersolution method with the comparison principle, yielding upper bounds for the blow-up time under two distinct parameter settings. As a special case, we derive the lower bound of the blow-up time when the boundary conditions transition from Robin to Neumann. The practicality and efficacy of our proposed methods are demonstrated through an example.

This paper focuses on exploring an impulsive stochastic differential variational inequality (ISDVI), which combines an impulsive stochastic differential equation and a stochastic variational inequality. Innovatively, our work incorporates two key aspects: first, our stochastic differential equation contains an impulsive term, enabling better handling of sudden event impacts; second, we utilize a non-local condition z(0)=χ0+ϑ(z) that integrates measurements from multiple locations to construct superior models. Methodologically, we commence our analysis by using the projection method, which ensures the existence and uniqueness of the solution to ISDVI. Subsequently, we showcase the practical applicability of our theoretical findings by implementing them in the investigation of a stochastic consumption process and electrical circuit model.

The stability for highly nonlinear impulsive coupled systems (HNICSs) with multi-weights is studied. It should be pointed out that the effects of impulsive disturbances are considered into highly nonlinear coupled systems. For HNICSs, existing impulsive differential inequalities cannot be used to obtain the stability criteria, which motivates us to construct a novel impulsive differential inequality and the inequality generalises the classic impulsive differential inequality to highly nonlinear cases. Furthermore, based on this inequality, Kirchhoff's Matrix Tree theorem in graph theory and the Lyapunov method, stability criteria are derived. Additionally, the coupled impulsive van der Pol-Duffing oscillators are presented. Correspondingly, the validity and practicability of the derived results are indicated with a numerical example.

We focus on Ulam type stability for fractional differential equations with nonlocal multi-point and multi-term boundary conditions. The application of Ulam stability to any type of boundary condition causes some misunderstandings which are mainly connected with the solutions of the applied inequality and the corresponding boundary condition. The main idea of Ulam type stability is the closeness between any solution of the corresponding differential inequality and the solution of the studied problem. Also, both solutions have to be deeply connected. To avoid misunderstandings in the literature we suggest three different approaches. In all of them we study the appropriately defined Ulam type stability.

Topological properties of solution sets for control problems driven by fractional delay differential quasi-hemivariational inequalities and applications

Abstract This article chronicles a development that started around 1990 with [17], where the authors showed that if a smooth bounded pseudoconvex domain $$\Omega $$ Ω in $$\mathbb {C}^{n}$$ C n admits a defining function that is plurisubharmonic at points of the boundary, then the $$\overline{\partial }$$ ∂ ¯ –Neumann operators on $$\Omega $$ Ω preserve the Sobolev spaces $$W^{s}_{(0,q)}(\Omega )$$ W ( 0 , q ) s ( Ω ) , $$s\ge 0$$ s ≥ 0 . The same authors then proved a further regularity result and made explicit the role of D’Angelo forms for regularity [19]. A few years later, Kohn [69] initiated a quantitative study of the results in [17] by relating the Sobolev level up to which regularity holds to the Diederich–Fornæss index of the domain. Many of these ideas were synthesized and developed further by Harrington [51–53]. Then, around 2020, Liu [72, 73] and Yum [101] discovered that the DF–index is closely related to certain differential inequalities involving D’Angelo forms. This relationship in turn led to a recent new result which supports the conjecture that DF–index one should imply global regularity in the $$\overline{\partial }$$ ∂ ¯ –Neumann problem [75]. Much of the work described above relies heavily on Kohn’s groundbreaking contributions to the regularity theory of the $$\overline{\partial }$$ ∂ ¯ –Neumann problem.