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Global Asymptotic Stability Research Articles

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4763 Articles

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Articles published on Global Asymptotic Stability

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A Delayed Malware Propagation Model Under a Distributed Patching Mechanism: Stability Analysis

Antivirus (patch) is one of the most powerful tools for defending against malware spread. Distributed patching is superior to its centralized counterpart in terms of significantly lower bandwidth requirement. Under the distributed patching mechanism, a novel malware propagation model with double delays and double saturation effects is proposed. The basic properties of the model are discussed. A pair of thresholds, i.e., the first threshold R0 and the second threshold R1, are determined. It is shown that (a) the model admits no malware-endemic equilibrium if R0≤1, (b) the model admits a unique patch-free malware-endemic equilibrium and admits no patch-endemic malware-endemic equilibrium if 1<R0≤R1, and (c) the model admits a unique patch-free malware-endemic equilibrium and a unique patch-endemic malware-endemic equilibrium if R0>R1. A criterion for the global asymptotic stability of the malware-free equilibrium is given. A pair of criteria for the local asymptotic stability of the patch-free malware-endemic equilibrium are presented. A pair of criteria for the local asymptotic stability of the patch-endemic malware-endemic equilibrium are derived. Using cybersecurity terms, these theoretical outcomes have the following explanations: (a) In the case where the first threshold can be kept below unity, the malware can be eradicated through distributed patching. (b) In the case where the first threshold can only be kept between unity and the second threshold, the patches may fail completely, and the malware cannot be eradicated through distributed patching. (c) In the case where the first threshold cannot be kept below the second threshold, the patches may work permanently, but the malware cannot be eradicated through distributed patching. The influence of the delays and the saturation effects on malware propagation is examined experimentally. The relevant conclusions reveal the way the delays and saturation effects modulate these outcomes.

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  • Journal IconMathematics
  • Publication Date IconJul 14, 2025
  • Author Icon Wei Zhang + 2
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Mode-dependent dwell time-based stabilisation of cyclic switched nonlinear systems using sum of squares programming

This paper addresses the stabilisation problem of switched nonlinear systems under cyclic switching functions with dwell time ranges and using sum-of-squares programming. We establish sufficient conditions for global uniform asymptotic and global uniform exponential stability of switched nonlinear systems with respect to cyclic switching rules with dwell time ranges. The proposed conditions are independent of the stability property of each subsystem. They can be applied to a wide range of switched systems with all stable, partial stable partial unstable, and all unstable modes allowing both stabilising and destabilising switching phenomena. Moreover, the proposed stability conditions are mode-dependent and provide a high degree of flexibility in control system design and yield less conservative results. We also introduce the notion of discretised time-varying polynomial Lyapunov functions as an extension of existing discretised time-varying quadratic Lyapunov functions. Using this tool, the sum of squares conditions are developed for the stabilisation of polynomial switched systems as an important subclass of switched nonlinear systems. Illustrative examples are proposed to demonstrate the effectiveness and superiority of the proposed theoretical results.

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  • Journal IconInternational Journal of Systems Science
  • Publication Date IconJul 12, 2025
  • Author Icon Mohammad Hejri
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Global Asymptotic Stability in Cubic Systems with a Nilpotent Point at the Origin and with An Invariant Straight Line

Global Asymptotic Stability in Cubic Systems with a Nilpotent Point at the Origin and with An Invariant Straight Line

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  • Journal IconQualitative Theory of Dynamical Systems
  • Publication Date IconJul 7, 2025
  • Author Icon Érika Diz-Pita + 2
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Dynamics of deterministic and stochastic SEIR models with dual epidemic hypothesis and log-normal standard Ornstein–Uhlenbeck process

In our life there are various infectious diseases which lead to negative effects on the economy. Therefore, it is highly important to develop mathematical models to describe the spread of the pandemic. In this study, we first develop a deterministic SEI1I2R epidemic model incorporating double epidemic hypothesis. We first verify that there exists a unique global positive solution of the system with any positive initial value, then we investigate the local and global asymptotical stability of the equilibria. Following that, we incorporate the inherent stochasticity of the virus spread by reinterpreting the previous SEIR-type epidemic model of infectious diseases as stochastic systems modelled via the log-normal standard Ornstein–Uhlenbeck process. After that we study the stochastic dynamics of the model in detail. More precisely, we first analyze the existence and uniqueness of the global solution of the system. Then by using the Markov semigroup theory, we establish sufficient criteria for the existence and uniqueness of an invariant probability measure of the system, which implies the persistence of the disease in a community. Especially, under the same conditions as the local asymptotical stability of the endemic equilibrium, we get the specific form of the local probability density function around the quasi-endemic equilibrium of the stochastic system. Meanwhile, we also verify that the invariant global probability density function can be approximated by the local probability density function as the noise intensity tends to zero. In addition, we give some sufficient conditions for elimination of the disease. Finally, numerical simulations are carried out to demonstrate the validity of the derived results and the availability of the proposed models by using MATLAB. This work can improve our insight into epidemic dynamics and help us implement interventions to regulate the disease dynamics.

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  • Journal IconJournal of Mathematical Physics
  • Publication Date IconJul 1, 2025
  • Author Icon Qun Liu + 1
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Dynamic Analysis of an Indirect Prey‐Taxis Model With Singular Sensitivity

ABSTRACTIn this paper, we consider an indirect prey‐taxis model with singular sensitivity. One of the main obstacles in the research is the possible singularity of the system. We first study the global existence of the unique classical solution of the system in a bounded convex region with smooth boundary and Neumann boundary conditions. We further investigate the global boundedness of the solutions. Then, by constructing some proper Lyapunov functionals, we show the global asymptotic stability of the steady states and give the rate of convergence of the solution. In addition, we discuss the local stability of the predator‐free steady state and positive constant steady state by using the corresponding characteristic equations. And adopting the indirect prey‐taxis coefficient as the bifurcation parameter, we analyze the occurrence of Hopf bifurcation and steady‐state bifurcation. Our results reveal that indirect prey‐taxis can destroy the stability, with higher chemotactic intensities making the system more likely to exhibit time‐periodic patterns, while lower chemotactic intensities make the system more likely to display steady‐state patterns. Among other things, we conduct a comparative analysis with the nonsingular indirect prey‐taxis system. Finally, several numerical simulations are presented to illustrate the findings.

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  • Journal IconStudies in Applied Mathematics
  • Publication Date IconJul 1, 2025
  • Author Icon Zhuzhen Liao + 1
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Finite-time robust state feedback control for flexible joint robotic manipulators in the presence of matched and mismatched uncertainties with experimental validation

This paper presents a finite-time robust controller for position-tracking control of flexible-joint robotic manipulators with n degrees of freedom in the presence of matched and mismatched uncertainties within a structured framework. First, the manipulator’s model is formulated in a state-space tracking form, considering uncertainties. Subsequently, a robust state-feedback controller is designed using a voltage-based control strategy. The proposed controller does not require precise knowledge of the system dynamics. This significantly reduces the complexity of the input control design. Moreover, its low computational demand makes the controller practical and cost-effective to implement. Mathematical analysis proves that the closed-loop system achieves finite-time global asymptotic stability under the proposed control despite uncertainties. Furthermore, the effectiveness of the proposed controller is demonstrated through quantitative and qualitative studies, including multi-scenario simulations and hardware-in-the-loop practical implementations. The results confirm the method’s superior performance, demonstrating reduced steady-state error, effective vibration suppression, lower energy consumption, and enhanced robustness.

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  • Journal IconInternational Journal of Control
  • Publication Date IconJun 28, 2025
  • Author Icon Abdollah Hasan Nezhad + 3
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Cascade-Based Distributed Estimator Tracking Control for Swarm of Multiple Nonholonomic Wheeled Mobile Robots via Leader–Follower Approach

This study aims to explore the tracking control challenge in a swarm of multiple nonholonomic wheeled mobile robots (NWMRs) by utilizing a distributed leader–follower strategy grounded in the cascade system theory. Firstly, the kinematic control law is developed for the leader by constructing a sliding surface based on the error tracking model with a virtual reference trajectory. Secondly, a communication topology with the desired formation pattern is modeled for the multiple robots by using the graph theory. Further, in the leader–follower NWMR system, each follower lacks direct access to the leader’s information. Therefore, a novel distributed-based controller by PD-based controller for the follower is developed, enabling each follower to obtain the leader’s information. Thirdly, for each case, we give a further analysis of the closed-loop system to guarantee uniform global asymptotic stability with the conditions based on the cascade system theory. Finally, the trajectory tracking performance of the proposed controllers for the NWMR system is illustrated through simulation results. The leader robot achieved a low RMSE of 1.6572 (Robot 1), indicating accurate trajectory tracking. Follower robots showed RMSEs of 2.6425 (Robot 2), 3.0132 (Robot 3), and 4.2132 (Robot 3), reflecting minor variations due to the distributed control strategy and local disturbances.

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  • Journal IconRobotics
  • Publication Date IconJun 26, 2025
  • Author Icon Dinesh Elayaperumal + 5
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Distributed Projection Neurodynamic Approaches in Continuous and Discrete Time for BP With Block Decomposition of Measurement Matrix.

Aiming at the situation where the measurement matrix B has a flexible block decomposition, this article designs two novel distributed continuous- and discrete-time projection neurodynamic approaches to solve the basis pursuit (BP) problem for sparse recovery. These approaches only require information from each flexible block of the measurement matrix B, rather than from each row, column, or the entire matrix. First, with the aid of the primal-dual dynamical approach, projection operator, and second-order multiagent consensus condition, a novel distributed projection neurodynamic approach in continuous time (DPNA-CT-B) is proposed, and its optimality and global asymptotic stability are rigorously proved. Moreover, based on the forward and backward Euler methods and variable substitution methods, a corresponding distributed projection neurodynamic approach in discrete time (DPNA-DT-B) is designed. Finally, through sparse signal and image reconstruction experiments, the effectiveness and superiority of the proposed neurodynamic approaches are verified.

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  • Journal IconIEEE transactions on neural networks and learning systems
  • Publication Date IconJun 23, 2025
  • Author Icon You Zhao + 4
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Geometric line-of-sight guidance law with exponential switching sliding mode control for marine vehicles’ path following

Marine vehicle guidance and control technology serves as the core support for advancing marine development and enabling scientific exploration. Its accuracy, autonomy, and environmental adaptability directly determine a vehicle’s mission effectiveness in complex marine environments. This paper explores path following for marine vehicles in the horizontal plane. To tackle the limitation of a fixed look-ahead distance, we develop a novel geometric line-of-sight guidance law. It adapts to diverse compound paths by dynamically adjusting according to cross-track errors and local path curvature. Then, to enhance control performance, we present an improved exponential switching law for sliding mode control, enabling rapid convergence, disturbance rejection, and chatter reduction. Additionally, integral sliding mode control is integrated to stabilize yaw angular velocity, ensuring the system’s global asymptotic stability. Through a series of numerical simulations, the effectiveness, robustness, and adaptability of our proposed methods are verified.

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  • Journal IconFrontiers in Robotics and AI
  • Publication Date IconJun 23, 2025
  • Author Icon Chengren Yuan + 5
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Secure Stabilization of Nonlinear Switched System via Practical Asynchronous Event‐Triggered Control

ABSTRACTIn this article, a practical asynchronous event‐triggered control (ETC) scheme and a strictly monotonically decreasing Lyapunov–Krasovskii functional (LKF) approach are proposed to study the secure global asymptotic stabilization (SGAS) and nonweighted performance for a nonlinear switched system (NSS) under deception attack. The proposed ETC is mode‐dependent with transmission delay in both the switching signal and states of the controller, which is more practical than previous asynchronous ETCs only considered transmission delay in the switching signal. The novelty lies in the realization that the LKF is strictly monotonically decreasing when both the transmission delay and the time‐varying delay exist, which can significantly decrease conservatism, and the nonweighted performance can be estimated with no extra conservative steps. Deriving solutions from linear matrix inequalities provides the controller gains. The reliability of the method is confirmed through numerical simulation and circuit simulation.

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  • Journal IconInternational Journal of Robust and Nonlinear Control
  • Publication Date IconJun 23, 2025
  • Author Icon Meijie Zhang + 3
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Analysis of Stability of Delayed Quaternion-Valued Switching Neural Networks via Symmetric Matrices

The stability of a class of quaternion-valued switching neural networks (QVSNNs) with time-varying delays is investigated in this paper. Limited prior research exists on the stability analysis of quaternion-valued neural networks (QVNNs). This paper addresses the stability analysis of quaternion-valued neural networks (QVNNs). With the help of some symmetric matrices with excellent properties, the stability analysis method in this paper is undecomposed. The QVSNN discussed herein evolves with average dwell time. Based on the Lyapunov theoretical framework and Wirtinger-based inequality, QVSNNs under any switching law have global asymptotic stability (GAS) and global exponential stability (GES). The state decay estimation of the system is also given and proved. Finally, the effective and practical applicability of the proposed method is demonstrated by two comprehensive numerical calculations.

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  • Journal IconSymmetry
  • Publication Date IconJun 20, 2025
  • Author Icon Yuan Dong + 4
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Asymptotic Stability of a Rumor Spreading Model with Three Time Delays and Two Saturation Functions

Time delays and saturation effects are critical elements describing complex rumor spreading behaviors. In this article, a rumor spreading model with three time delays and two saturation functions is proposed. The basic properties of the model are reported. The structure of the rumor-endemic equilibria is deduced. A criterion for the global asymptotic stability of the rumor-free equilibrium is derived. In the presence of very small delays, a criterion for the local asymptotic stability of a rumor-endemic equilibrium is provided. The influence of the delays and the saturation effects on the dynamics of the model is made clear through simulation experiments. In particular, it is found that (a) extended time delays lead to slower change in the number of spreaders or stiflers and (b) lifted saturation coefficients lead to slower change in the number of spreaders or stiflers. This work helps to deepen the understanding of complex rumor spreading phenomenon and develop effective rumor-containing schemes.

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  • Journal IconMathematics
  • Publication Date IconJun 18, 2025
  • Author Icon Teng Sheng + 4
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Switching linear parameter varying control for a robotic manipulator

This paper introduces a synthesis study of a switching Linear Parameter Varying (LPV) controller with multiple polytopic subregions, considering separate parameter-dependent Lyapunov functions. The parameter space setup is designed with overlapping subregions in which safe controller switching is ensured by hysteresis switching logic. Robotic manipulators, like any mechatronic systems, are burdened with unmodelled characteristics such as friction phenomena and actuator dynamics, as well as nonlinear dynamics. These effects might be countered by LPV controllers, although they may be too conservative for production. The switching controller can decrease conservatism utilising separate Lyapunov functions while maintaining essential features of a general H∞ LPV controller, such as global asymptotic stability over the entire parameter space. The suggested method is applied on a real-life 2-DOF SCARA-type robotic manipulator, and extensive testing results are presented along with some of the key challenges that arise when the application is moved from simulation to real-life. Primary considerations with respect to the application of switching control are studied in detail, including the number and shape of subregions, overlapping percentage, switching surface orientation, and scheduling parameter selection.

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  • Journal IconActa IMEKO
  • Publication Date IconJun 17, 2025
  • Author Icon Zoltán Téczely + 1
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Stability Analysis of a k-Delayed Rational Difference Equation with Exponential Response

In this paper, we study the asymptotic and boundedness behavior of the solutions of the difference equation xn+1 = αe−xn + βxn−1 (1) A + Bxn + Cxn−k n = 0, 1, 2, . . . , where α, β, A, B, C are positive real numbers, k ∈ {1, 2, . . . }, and initial conditions x−k , x−k+1, . . . , x−1, x0 are arbitrary positive real numbers. Key words: Equilibrium point, Local asymptotic stability, Global asymptotic stability, Positive solutions, Boundedness, Periodicity.

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  • Journal IconJournal of Computational Mathemtica
  • Publication Date IconJun 16, 2025
  • Author Icon Sibi C Babu
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Asymptotical Behavior of Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Nonlinear Incidence Rates and Proportional Impulsive Vaccination

This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive SIR system is positive for arbitrary step size when the initial values are positive. By applying discrete Floquet’s theorem and small-amplitude perturbation skills, we proved that the disease-free periodic solution of the impulsive system is locally stable. Additionally, in conjunction with the discrete impulsive comparison theorem, we show that the impulsive linearly implicit Euler method maintains the global asymptotical stability of the exact solution of the impulsive system. Two numerical examples are provided to illustrate the correctness of the results.

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  • Journal IconAxioms
  • Publication Date IconJun 16, 2025
  • Author Icon Zhi-Wei Xu + 1
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Complex Dynamical Analysis of Two Prey‐One Predator Model in a Patch Environment Utilizing Spectral Bound

ABSTRACTIn this article, a two prey‐one predator model in which prey and predator disperse simultaneously in a heterogeneous environment with patches is proposed and analyzed. We prove that the solution of the system is positive and uniformly ultimately bounded. Meanwhile, we use the monotonic theory of spectral bounds to investigate the effect of the dispersal rate on population dynamics. To be precise, we discuss the stability behavior for the trivial equilibrium and semitrivial equilibrium as well as the uniform persistence of the system. Furthermore, we prove the global asymptotic stability of the positive equilibrium by constructing a global Lyapunov function which applies the results from graph theory. Some numerical simulations are provided to show the effectiveness of the theoretical results.

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  • Journal IconMathematical Methods in the Applied Sciences
  • Publication Date IconJun 13, 2025
  • Author Icon Min Zhang + 2
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Prescribed-time non-overshooting tracking control for nonlinear systems via a time-scale transformation approach*

This paper delves into addressing the issue of achieving non-overshooting tracking control within a prescribed time for a class of nonlinear systems. By employing a blow-up function and a time-scale transformation approach, the tracking error system is converted from the original time domain into a scaled system within the transformed time framework. Subsequently, a switching controller is then designed to ensure prescribed-time stability for the original tracking error system, which is achieved by applying a backstepping technique to analyze the global asymptotic stability of the scaled system. Additionally, conditions on the initial states of the tracking error system are provided to ensure the non-overshooting performance of nonlinear systems. Recommendations for selecting controller gains are also derived to guarantee non-overshooting prescribed-time stability under arbitrary initial conditions. Finally, the effectiveness of the proposed method is demonstrated through its application to a wheeled mobile robot system.

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  • Journal IconInternational Journal of Systems Science
  • Publication Date IconJun 10, 2025
  • Author Icon Mengnan Hao + 2
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Nonlinear Integral Extension of PID Control With Improved Convergence of Perturbed Second‐Order Dynamic Systems

ABSTRACTNonlinear extension of the integral part of a standard proportional‐integral‐derivative (PID) feedback control is proposed for perturbed second‐order systems. The approach is model‐free and requires solely the Lipschitz boundedness of the unknown matched perturbations. For constant disturbances, the global asymptotic stability is shown based on the circle criterion. For Lipschitz perturbations, an ultimately bounded output error is provided based on the steady‐state behavior in frequency domain. Also the transient response to the stepwise disturbances is analyzed for the control tuning. Based on the developed analysis, the design recommendations are formulated as a step‐by‐step procedure. It is also discussed how the proposed control is applicable to second‐order systems extended by additional (parasitic) actuator dynamics with low‐pass characteristics. The proposed nonlinear control is proven to outperform its linear PID counterpart during the settling phase, that is, at convergence of the residual output error. An experimental case study of the second‐order system with an additional actuator dynamics and considerable perturbations is demonstrated to confirm and benchmark the control performance.

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  • Journal IconAdvanced Control for Applications
  • Publication Date IconJun 1, 2025
  • Author Icon Michael Ruderman
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Dynamics and asymptotic profiles of a local-nonlocal dispersal SIR epidemic model with spatial heterogeneity.

Dynamics and asymptotic profiles of a local-nonlocal dispersal SIR epidemic model with spatial heterogeneity.

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  • Journal IconInfectious Disease Modelling
  • Publication Date IconJun 1, 2025
  • Author Icon Salih Djilali + 2
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A mathematical model for analyzing the spread of malaria

This paper examines the dynamics of a disease transmission model by estimating the basic reproduction number [Formula: see text] using the next-generation matrix approach. The stability of the disease-free equilibrium is studied, revealing global asymptotic stability when [Formula: see text], suggesting mosquito extinction, and instability when [Formula: see text], indicating mosquito persistence. The criteria for the emergence of a distinct endemic equilibrium are identified, and the stability of both disease-free and endemic equilibria is investigated locally and globally. Numerical simulations are used to evaluate the analytical results and examine the model’s probable behavior under different scenarios.

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  • Journal IconInternational Journal of Biomathematics
  • Publication Date IconMay 30, 2025
  • Author Icon Gizachew Tirite Gellow + 1
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