Global Stability Research Articles

Global existence and stability of the wave equation with boundary variable damping

AbstractIn this paper, we present the result of global existence of solution for the wave equation with boundary variable damping term. Then, we prove that this global solution is stable. Our study is based on the semi‐groups theory and some integral inequalities.

Reducing the number and the size of sovereign defaults in the world: Long-term trend or temporary phenomenon?

The paper explains the current global trends, which consist in a decrease in both the number and the size of defaults. For this purpose, a new approach to modeling the probability and the size of defaults based on nonlinear models (generalized additive models) has been applied. The obtained empirical estimates confirm the significant nonlinearity of the relationships between factors. Based on the proposed approach, it is possible to detect the disciplinary effect of the rate of increase in interest expenses and confirm the significance of the rate of debt accumulation as a development of previous works. We have found that the main contribution to this trend is made by the institutional factor — government efficiency: the more effective the government is, the better it ensures debt sustainability, reducing both the number of defaults and their size. Another important factor is the soft monetary policy of developed countries. It affects the default size, reducing it, but is not the reason for the decrease in the number (probability) of defaults in the world. Thus, the current trend is long-term and shows the progress of some countries and international organizations in reducing risks to global financial stability.

Global Stability of SIR Model with Heterogeneous Transmission Rate Modeled by the Preisach Operator

Global Stability of SIR Model with Heterogeneous Transmission Rate Modeled by the Preisach Operator

Study on SEAI Model of COVID-19 Based on Asymptomatic Infection

In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number R0 and calculate the equilibrium point. Secondly, when R0<1, the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When R0>1, the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of R0 is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease.

Global stability analysis and modelling onchocerciasis transmission dynamics with control measures

ABSTRACT Background: Onchocerciasis infection is one of the neglected tropical diseases targeted for eradication by 2030. The disease is usually transmitted to humans through the bites of black flies. These black flies mostly breed near well-oxygenated fast-running water bodies. The disease is common in mostly remote agricultural villages near rivers and streams. Objective: In this study, a deterministic model describing the infection dynamics of human onchocerciasis disease with control measures is presented. Methods: We derived the model’s reproductive number and used a stability theorem of a Metzler matrix to show that disease-free equilibrium is both locally and globally asymptotically stable whenever the reproductive number is less than one. Parameter contribution was conducted using sensitivity analysis. The model endemic equation is shown to be a cubic polynomial in the presence of infected immigrants and a quadratic form in their absence. Results: When the inflow of infected immigrants is null, the model endemic equation may admit a unique equilibrium if the reproductive number is greater than one, or admits multiple endemic equilibria if the reproductive number is less than unity. We carried out a sensitivity analysis to identify the significant parameters that contribute to onchocerciasis spread. Conclusion: Onchocerciasis disease can be eradicated if the importation of infected immigrants is properly monitored. The integration of the One Health concept in the public health system is key in tackling the emergence and spread of diseases.

Role of Allee and Fear for Controlling Chaos in a Predator–Prey System with Circulation of Disease in Predator

This paper explores a predator–prey system featuring fear and disease within the predator population,utilizing the Rosenzweig–MacArthur model with Holling type-II functional response. The primary focus lies in investigating the impact of a fear factor, wherein the prey’s growth rate is hindered due to predator-induced fear. Additionally, the model accounts for the spread of disease among predators,leading to a division between susceptible and infected predator subpopulations. The inclusion of an Allee effect in the susceptible predator further enriches the model. The study involves a thorough examination, encompassing local and global stability analysis as well as Hopf bifurcation analysis around the interior equilibrium point. Numerical simulations underscore a noteworthy observation: an escalation in interaction force propels the system into chaotic dynamics,marked by stable focus, limit cycles and period-doubling phenomena. A noteworthy finding pertains to the influence of the Allee parameter ([Formula: see text]) on chaotic dynamics. As the Allee parameter values increase, the system tends to stable focus through a sequence of chaotic states, period-doubling and limit cycles. Subsequently, the paper introduces the role of another pivotal parameter, the fear factor, into the chaotic dynamics. Intriguingly, chaos transforms into stable focus through diverse nonlinear phenomena, including period-doubling and limit cycles. This nuanced exploration of parameters sheds light on the intricate dynamics governing the predator–prey system, offering a comprehensive understanding of the interplay between fear, disease and ecological factors. So our observation throughout this paper that how chaos behaves here after one by one injection of our new features: fear factor and Allee parameter?

Russia’s Declining Satellite Reconnaissance Capabilities and Its Implications for Security and International Stability

Satellite-based intelligence has played a crucial role in national security and international stability. This article explores the persistent decline in Russia’s satellite reconnaissance capabilities since the Cold War’s end, challenging hopes for recovery and highlighting systemic issues. It delves into the impact on Russia’s military actions in Ukraine, where the absence of satellites contributes to indiscriminate shelling and higher military casualties. Furthermore, it examines the repercussions on nuclear arms control and the peaceful use of space, critical for global stability. As Russia struggles with reconnaissance gaps, its pursuit of antisatellite weapons poses a threat to the principle of space for peace. This analysis aims to deepen insights into the intelligence decline during the Russia–Ukraine conflict and its potential ramifications for international stability.

Mathematical modeling and control of Covid‐19

In this paper, the author proposes and studies a mathematical model for the spread of coronavirus 2019. The effect of preventive programs on the control of disease is also considered. The basic reproduction number was computed and the results showed that forward bifurcation occured. Local and global stability of the equilibrium points of the model was studied by geometric stability method and Lyapunov function. Finally, the particular data of China which was reported for Covid‐19 were used for numerical simulation.

Effects of stochastic perturbations on a phytoplankton allelopathy competitive system

In this paper, we investigate stochastic models of two species competitive phytoplankton system with allelopathic interaction. We construct stochastic models from deterministic models by introducing three different stochastic perturbations to the equations. For the first model, we prove that this model is asymptotically stable in probability at three different equilibrium points. It is also shown that the system is positively recurrent under some conditions. For the second model, we study the extinction, existence of positive recurrence and global asymptotic stability under some sufficient conditions. For the last system, almost sufficient and necessary conditions for global asymptotic stability, the existence of positive recurrence and extinction of each population are established. In addition, using a suitable stochastic Lyapunov method, we prove that there is a unique solution to three models, we also discuss the ultimate boundedness of the three stochastic systems. Moreover, some numerical simulations are introduced to illustrate our mathematical results. We show that stochastic models still retain the desirable stability of their deterministic counterparts if stochastic perturbations are relatively small.

Existence and exponential stability of a periodic solution of an infinite delay differential system with applications to Cohen–Grossberg neural networks

In the present paper, we investigate both the global exponential stability and the existence of a periodic solution of a general differential equation with unbounded distributed delays. The main stability criterion depends on the dominance of the non-delay terms over the delay terms. The criterion for the existence of a periodic solution is obtained with the application of the coincidence degree theorem. We use the main results to get criteria for the existence and global exponential stability of periodic solutions of a generalized higher-order periodic Cohen–Grossberg neural network model with discrete-time varying delays and infinite distributed delays. Additionally, we provide a comparison with the results in the literature and two numerical examples to illustrate the effectiveness of some of our results.

Green finance and food production: Evidence from cities in China

In the domain of global socio-economic dynamics, the pivotal role of food production (FP) cannot be overstated. It constitutes the bedrock of fundamental human necessities while being inexorably linked with critical aspects such as economic expansion, societal equilibrium, and geopolitical interplays. The multifaceted nature of FP appears through its direct impact on macroeconomic indicators and its latent influence on international relations and policy-making. This critical nexus underscores the imperative for a comprehensive understanding and strategic management of food production processes to sustain and enhance global socio-economic stability and growth. Investigating the nexus between green finance (GF) and FP is pivotal for ensuring national security and promoting sustainable economic development. This research thus paper examines GF's impact on FP in China as outlined in the 2017 green finance policies. It employs the difference-in-differences (DID) method and analyzes a balanced panel dataset comprising 282 cities in China from 2009 to 2020. The results show that GF has an inhibitory effect on FP, and this remains valid after a range of reliability checks, including parallel trend assessments and placebo tests. Further findings of transmission mechanism analysis indicate that non-agricultural factors and financial exclusion are critical for GF's inhibitory effect on FP. Heterogeneity analyses show that GF's inhibitory effect on FP is more pronounced in the main food-producing areas, the main food-marketing areas, and cities with better infrastructure. The empirical results of this paper comprehensively illustrate GF's impact on FP and how it contributes to the achievement of food security objectives.

The role of taxation in an integrated economic-environmental model: a dynamical analysis

AbstractWe propose a model with economic and environmental domains that interact with each other. The economic sphere is described by a Solow growth model, in which productivity is not exogenous but negatively affected by the stock of pollution that stems from the production process. A regulator can charge a tax on production, and the resources collected from taxation are used to reduce pollution. The resulting model consists of a two dimensional discrete dynamical system, and we study the role of taxation from both a static and a dynamical point of view. The focus is on the determination of the conditions under which taxation has a positive effect on the environment and leads to economic growth. Moreover, we show that a suitable environmental policy can allow recovering both local and global stability of the steady states. On the contrary, we show that, if the policy is not adequate, the system can exhibit endogenous oscillating and chaotic behavior and multistability phenomena.

Modeling the Transmission Dynamics of Tuberculosis in Humans

Tuberculosis (TB) is a major threat to human health particularly in most of developing countries. In this article, we formulate and analyze a deterministic model for the transmission dynamics of pulmonary and extra-pulmonary tuberculosis. The next generation method is employed to find the basic reproduction number R0 which helps to determine whether TB clears or persists in the human population. Global stability of model equilibria is done through Lyapunov functions whereas the normalized forward sensitivity index method is adopted to determine parameters that drive tuberculosis. Analysis shows that both TB free and endemic equilibria exist. The TB free equilibrium is globally asymptotically stable whenever the basic reproduction number R0 < 1 whereas the endemic equilibrium is globally asymptotically stable whenever R0 > 1. Sensitivity analysis shows that the TB infection rate, the fraction of individuals who progress to pulmonary tuberculosis and its induced death drive TB. Numerical results indicate that when there are no interventions, susceptible humans decline significantly with time until when they are attracted to the steady state whereas latently infected, pulmonary and extra-pulmonary TB individuals increase until when they settle at the equilibrium states supporting the analytical results for existence of the endemic equilibrium. In light of these findings, we recommend treating humans infected with pulmonary TB who are carriers of the disease.

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Purpose To provide high torques needed to move a robot’s links, electric actuators are followed by a transmission system with a high transmission rate. For instance, gear ratios of 100:1 are often used in the joints of a robotic manipulator. This results into an actuator with large mechanical impedance (also known as nonback-drivable actuator). This in turn generates high contact forces when collision of the robotic mechanism occur and can cause humans’ injury. Another disadvantage of electric actuators is that they can exhibit overheating when constant torques have to be provided. Comparing to electric actuators, pneumatic actuators have promising properties for robotic applications, due to their low weight, simple mechanical design, low cost and good power-to-weight ratio. Electropneumatically actuated robots usually have better friction properties. Moreover, because of low mechanical impedance, pneumatic robots can provide moderate interaction forces which is important for robotic surgery and rehabilitation tasks. Pneumatic actuators are also well suited for exoskeleton robots. Actuation in exoskeletons should have a fast and accurate response. While electric motors come against high mechanical impedance and the risk of causing injuries, pneumatic actuators exhibit forces and torques which stay within moderate variation ranges. Besides, unlike direct current electric motors, pneumatic actuators have an improved weight-to-power ratio and avoid overheating problems. Design/methodology/approach The aim of this paper is to analyze a nonlinear optimal control method for electropneumatically actuated robots. A two-link robotic exoskeleton with electropneumatic actuators is considered as a case study. The associated nonlinear and multivariable state-space model is formulated and its differential flatness properties are proven. The dynamic model of the electropneumatic robot is linearized at each sampling instance with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. Within each sampling period, the time-varying linearization point is defined by the present value of the robot’s state vector and by the last sampled value of the control inputs vector. An H-infinity controller is designed for the linearized model of the robot aiming at solving the related optimal control problem under model uncertainties and external perturbations. An algebraic Riccati equation is solved at each time-step of the control method to obtain the stabilizing feedback gains of the H-infinity controller. Through Lyapunov stability analysis, it is proven that the robot’s control scheme satisfies the H-infinity tracking performance conditions which indicate the robustness properties of the control method. Moreover, global asymptotic stability is proven for the control loop. The method achieves fast convergence of the robot’s state variables to the associated reference trajectories, and despite strong nonlinearities in the robot’s dynamics, it keeps moderate the variations of the control inputs. Findings In this paper, a novel solution has been proposed for the nonlinear optimal control problem of robotic exoskeletons with electropneumatic actuators. As a case study, the dynamic model of a two-link lower-limb robotic exoskeleton with electropneumatic actuators has been considered. The dynamic model of this robotic system undergoes first approximate linearization at each iteration of the control algorithm around a temporary operating point. Within each sampling period, this linearization point is defined by the present value of the robot’s state vector and by the last sampled value of the control inputs vector. The linearization process relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. The modeling error which is due to the truncation of higher-order terms from the Taylor series is considered to be a perturbation which is asymptotically compensated by the robustness of the control algorithm. To stabilize the dynamics of the electropneumatically actuated robot and to achieve precise tracking of reference setpoints, an H-infinity (optimal) feedback controller is designed. Actually, the proposed H-infinity controller for the model of the two-link electropneumatically actuated exoskeleton achieves the solution of the associated optimal control problem under model uncertainty and external disturbances. This controller implements a min-max differential game taking place between: (i) the control inputs which try to minimize a cost function which comprises a quadratic term of the state vector’s tracking error and (ii) the model uncertainty and perturbation inputs which try to maximize this cost function. To select the stabilizing feedback gains of this H-infinity controller, an algebraic Riccati equation is being repetitively solved at each time-step of the control method. The global stability properties of the H-infinity control scheme are proven through Lyapunov analysis. Research limitations/implications Pneumatic actuators are characterized by high nonlinearities which are due to air compressibility, thermodynamics and valves behavior and thus pneumatic robots require elaborated nonlinear control schemes to ensure their fast and precise positioning. Among the control methods which have been applied to pneumatic robots, one can distinguish differential geometric approaches (Lie algebra-based control, differential flatness theory-based control, nonlinear model predictive control [NMPC], sliding-mode control, backstepping control and multiple models-based fuzzy control). Treating nonlinearities and fault tolerance issues in the control problem of robotic manipulators with electropneumatic actuators has been a nontrivial task. Practical implications The novelty of the proposed control method is outlined as follows: preceding results on the use of H-infinity control to nonlinear dynamical systems were limited to the case of affine-in-the-input systems with drift-only dynamics. These results considered that the control inputs gain matrix is not dependent on the values of the system’s state vector. Moreover, in these approaches the linearization was performed around points of the desirable trajectory, whereas in the present paper’s control method the linearization points are related with the value of the state vector at each sampling instance as well as with the last sampled value of the control inputs vector. The Riccati equation which has been proposed for computing the feedback gains of the controller is novel, so is the presented global stability proof through Lyapunov analysis. This paper’s scientific contribution is summarized as follows: (i) the presented nonlinear optimal control method has improved or equally satisfactory performance when compared against other nonlinear control schemes that one can consider for the dynamic model of robots with electropneumatic actuators (such as Lie algebra-based control, differential flatness theory-based control, nonlinear model-based predictive control, sliding-mode control and backstepping control), (ii) it achieves fast and accurate tracking of all reference setpoints, (iii) despite strong nonlinearities in the dynamic model of the robot, it keeps moderate the variations of the control inputs and (iv) unlike the aforementioned alternative control approaches, this paper’s method is the only one that achieves solution of the optimal control problem for electropneumatic robots. Social implications The use of electropneumatic actuation in robots exhibits certain advantages. These can be the improved weight-to-power ratio, the lower mechanical impedance and the avoidance of overheating. At the same time, precise positioning and accurate execution of tasks by electropneumatic robots requires the application of elaborated nonlinear control methods. In this paper, a new nonlinear optimal control method has been developed for electropneumatically actuated robots and has been specifically applied to the dynamic model of a two-link robotic exoskeleton. The benefit from using this paper’s results in industrial and biomedical applications is apparent. Originality/value A comparison of the proposed nonlinear optimal (H-infinity) control method against other linear and nonlinear control schemes for electropneumatically actuated robots shows the following: (1) Unlike global linearization-based control approaches, such as Lie algebra-based control and differential flatness theory-based control, the optimal control approach does not rely on complicated transformations (diffeomorphisms) of the system’s state variables. Besides, the computed control inputs are applied directly on the initial nonlinear model of the electropneumatic robot and not on its linearized equivalent. The inverse transformations which are met in global linearization-based control are avoided and consequently one does not come against the related singularity problems. (2) Unlike model predictive control (MPC) and NMPC, the proposed control method is of proven global stability. It is known that MPC is a linear control approach that if applied to the nonlinear dynamics of the electropneumatic robot, the stability of the control loop will be lost. Besides, in NMPC the convergence of its iterative search for an optimum depends on initialization and parameter values selection and consequently the global stability of this control method cannot be always assured. (3) Unlike sliding-mode control and backstepping control, the proposed optimal control method does not require the state-space description of the system to be found in a specific form. About sliding-mode control, it is known that when the controlled system is not found in the input-output linearized form the definition of the sliding surface can be an intuitive procedure. About backstepping control, it is known that it cannot be directly applied to a dynamical system if the related state-space model is not found in the triangular (backstepping integral) form. (4) Unlike PID control, the proposed nonlinear optimal control method is of proven global stability, the selection of the controller’s parameters does not rely on a heuristic tuning procedure, and the stability of the control loop is assured in the case of changes of operating points. (5) Unlike multiple local models-based control, the nonlinear optimal control method uses only one linearization point and needs the solution of only one Riccati equation so as to compute the stabilizing feedback gains of the controller. Consequently, in terms of computation load the proposed control method for the electropneumatic actuator’s dynamics is much more efficient.

Mapping annual 10-m soybean cropland with spatiotemporal sample migration

China, as the world’s biggest soybean importer and fourth-largest producer, needs accurate mapping of its planting areas for global food supply stability. The challenge lies in gathering and collating ground survey data for different crops. We proposed a spatiotemporal migration method leveraging vegetation indices’ temporal characteristics. This method uses a feature space of six integrals from the crops’ phenological curves and a concavity-convexity index to distinguish soybean and non-soybean samples in cropland. Using a limited number of actual samples and our method, we extracted features from optical time-series images throughout the soybean growing season. The cloud and rain-affected data were supplemented with SAR data. We then used the random forest algorithm for classification. Consequently, we developed the 10-meter resolution ChinaSoybean10 maps for the ten primary soybean-producing provinces from 2019 to 2022. The map showed an overall accuracy of about 93%, aligning significantly with the statistical yearbook data, confirming its reliability. This research aids soybean growth monitoring, yield estimation, strategy development, resource management, and food scarcity mitigation, and promotes sustainable agriculture.

Spatiotemporal dynamics of prey-predator model incorporating Holling-type II functional response with fear and its carryover effects.

The recent focus in the fields of biology and ecology has centered on the significant attention given to the mathematical modeling and analyzing the spatiotemporal population distribution among species engaged in interactions. This paper explores the dynamics of the temporal and spatiotemporal delayed Bazykin-type prey-predator model, incorporating fear and its carryover effect. In our model, we incorporated a functional response of the Holling-type II. In the temporal model, a detailed dynamic analysis was carried out, investigating the positivity and boundedness of solutions, establishing the uniqueness and existence of positive interior equilibria, and examining both local and global stability. Additionally, we explored the presence of saddle-node, transcritical, and Hopf bifurcations varying attack rate parameter. The delayed system shows highly periodic behavior. Additionally, for the spatiotemporal model, we provide a complete analysis of local and global stability, and we derive the conditions for the existence of Turing instability for both self-diffusion and cross-diffusion, respectively. The two-dimensional diffusive model is further discussed, highlighting various Turing patterns, including holes, stripes, and hot and cold spots, along with their biological significance. Numerical simulations are executed to validate the analytical findings in both temporal and spatiotemporal models.

An intuitionistic fuzzy approach for prey-predator harvesting system with toxicity and time delay

This study presents an intuitionistic fuzzy prey-predator model with time delay, where several toxic substances from external sources contaminate the prey and predator species. The system consists of one prey and one predator, with the logistic growth of prey species and predation terms proportional to prey density. This system incorporates Holling type-II functional response and introduces a time delay in harvesting prey species over a precise size or age. The primary focus of this work is to investigate the impact of age or size-selective delay on the system’s stability within the intuitionistic fuzzy environment. The model employs triangular intuitionistic fuzzy numbers to account for imprecise biological parameters, and it is converted into an intuitionistic fuzzy model using the Hukuhara derivative. The fuzziness of the model is then resolved using Yager’s Ranking method. Without delay, the paper delves into the system’s dynamics, examining local and global stability and bionomic considerations. Numerical simulations are performed using MATLAB to validate the theoretical findings.

Fractional mathematical modelling of breast cancer stages with true data from Saudi Arabia

This research focuses on the development, analysis, and simulation of fractional mathematical models to investigate the transmission dynamics of different phases of breast cancer. The suggested breast cancer model incorporates three often-used fractional operators in epidemiology: Caputo, Caputo–Fabrizio-Caputo, and Atangana-Baleanu-Caputo operators. In this study, the determination of the equilibrium point and its stability analysis is conducted using the Routh–Hurwitz criterion. Additionally, we examine the existence and uniqueness of solutions for the fractional system using Krasnoselskii’s and Banach fixed-point theory. Moreover, the global stability is discussed via the Ulam-Hyres criterion. Furthermore, the fractional models are being verified using reported occurrences of stage IV breast cancer among females in Saudi Arabia from 2004 to 2016. The real data is used to determine the values of the parameters that are fitted using the least squares error-minimizing methodology. Also, residuals and efficiency rates are computed for the integer as well as fractional-order models. Graphical representations are used to illustrate numerical results by examining different choices of fractional order parameters. Then, the dynamic characteristics of various stages of breast cancer are analyzed to demonstrate the impact of fractional order on breast cancer progression and how the rate of chemotherapy influences its behavior. We provide graphical results for a breast cancer model with effective parameters, resulting in fewer future incidences in the population of stages III and IV. Chemotherapy often raises the risk of cardiotoxicity, and our proposed model output reflects this. The goal of this study is to reduce the incidence of cardiotoxicity in chemotherapy patients while also increasing the pace of patient recovery. This research has the potential to significantly improve outcomes for patients and provide information on treatment strategies for breast cancer patients.

Cadmium(II) adsorption by recyclable Zeolite-Loaded Hydrogel: Extension to the removal of Cadmium(II) from contaminated soil

Excessive cadmium(II) (Cd(II)) in soil has gravely jeopardized global ecosystem stability and human health. The removal of Cd(II) is a permanent cure to remediate Cd(II)-contaminated soil. Herein, a recyclable removal material, zeolite@cellulose-poly(acrylamide) hydrogel (Z@CA) was developed. Batch adsorption experiments and characterisations (SEM-EDS, FTIR, XRD, and XPS) were conducted to explore the adsorption capacity and mechanism of Z@CA. A pot experiment was established to evaluate the effect of Z@CA on soil Cd(II) remediation and Cd(II) toxicity in water spinach. The results showed that in aqueous solutions, the maximum adsorption capacity of Cd(II) by Z@CA was 84.4 mg·g−1. Electrostatic attraction, ion exchange, and surface complexation collectively facilitated Cd(II) capture by Z@CA. Even after five adsorption–desorption cycles, Z@CA maintained a Cd(II) removal efficiency of 89.88 %. The results of the pot experiment showed that after applying Z@CA (2.5 w/w) in soil for 21 days, the content of bioavailable Cd(II) and total Cd(II) significantly decreased by 59.38 % and 1.75 %, respectively. The recovery rate of Z@CA reached 94.77 % and maintained its original intact structure. In addition, the Cd(II) content of water spinach was reduced by 91.43 % compared to the treatment without Z@CA, the weight, height, and relative chlorophyll content reaching 1.34, 1.73, and 1.49 times that of the treatment without Z@CA. The ideal application of Z@CA provides novel approaches and insights into the remediation of Cd(II)-contaminated soils.

From Aerial to underwater: Modeling and trajectory tracking control of the underactuated quadrotor hybrid Aerial underwater vehicle

The underactuated Quadrotor Hybrid Aerial Underwater Vehicle (HAUV) can operate in multiple mediums and seamlessly transition between them. Affected by the buoyancy, underwater control suffers a critical 180° turnover singular problem. To address this challenge, this paper presents a detailed underwater 6-degree of freedom dynamics based on the unit quaternions. A novel geometric attitude extraction method is introduced to improve the classical geometric control. Subsequently, a cascaded controller is developed for trajectory tracking. This controller consists of a position loop and an attitude loop, integrated through the attitude extraction method. It extracts the desired attitude for the inner loop from the desired thrust direction in the outer loop. Rigorous analysis demonstrates that the closed system exhibits global asymptotic stability, effectively mitigating the singular problem. Simulations demonstrate that the controller effectively adapts the quadrotor HAUV, regardless of its buoyancy (positive, negative, or neutral), to accurately track underwater trajectories and fixed points.