Stability Theory Research Articles

This paper addresses the secure synchronisation of uncertain complex networks (UCNs) under stochastic deception attacks (DAs) using a novel impulsive control strategy. We focus on DAs targeting sensor-to-controller communication channels, where attackers inject distinct false data with varying attack probabilities across different channels. To counter these attacks, we propose a novel secure impulsive controller and analyze the resulting bounded synchronisation dynamics. Through the integration of Lyapunov stability theory and average impulsive interval (AII) analysis, we establish sufficient conditions for achieving mean-square bounded synchronisation and derive the corresponding synchronisation bounds. The theoretical framework is comprehensively validated through numerical simulations, which demonstrate the efficacy of our proposed control approach in maintaining system performance under complex attack scenarios.

This study presents a comprehensive Lyapunov-based framework for analyzing partial practical stability in nonlinear tempered fractional-order systems (TFOS). We develop novel stability concepts including β*-practical uniform generalized Mittag–Leffler stability (β*-PUGMLS) and β*-practical uniform exponential stability (β*-PUES) with respect to system substates. Through carefully constructed Lyapunov functions, we establish sufficient conditions under which the system’s states converge to a predefined neighborhood of the origin. The theoretical framework provides Mittag–Leffler and exponential stability criteria for tempered fractional-order systems, extending classical stability theory to this important class of systems. Furthermore, we apply these stability results to design stabilizing feedback controllers for a specific class of triangular TFOS, demonstrating the practical utility of our theoretical developments. The efficacy of the proposed stability criteria and control strategy is validated through several illustrative examples, showing that system states converge appropriately under the derived conditions. This work contributes significantly to the stability theory of fractional-order systems and provides practical tools for controlling complex nonlinear systems in the tempered fractional calculus framework.

The vibration reduction performance of NES is greatly affected by the excitation. When the excitation is too high, the system may experience bifurcation, leading to the vibration reduction failure of NES. For the above issues, a nonlinear damping parallel energy sink (NDPNES) is proposed in this paper, which utilizes damping to increase with the increase of relative motion between the NES and the controlled system to ensure efficient vibration reduction performance under strong excitation. At the same time, without increasing the total mass, to improve the stability of the NES through mass configuration. Firstly, the mathematical model of NDPNES was established, and the slowly varying equation of the system was obtained based on the complex variable averaging (CX-A) method. Then, the amplitude frequency characteristics of the controlled system were obtained, and the stability was analyzed using Lyapunov stability theorem, at the same time, the analytical results were verified by numerical methods. Afterwards, the influence of parameters on the vibration reduction performance of NDPNES was analyzed. Finally, the performance of the proposed scheme under transient and steady-state excitation was verified using numerical methods. The results indicate that, compared to existing Cubic stiffness NES (CNES), the proposed NDPNES has better vibration reduction performance and stability under strong excitation. The proposed scheme is of great significance for vibration control of machinery (such as mining machinery, engineering machinery, etc.) operating in strong excitation environments.

Well-Posedness is essential in various scientific and engineering fields, including physics, engineering, biological sciences, economics, and environmental science. Well-posedness ensures that the mathematical model corresponds to a physically meaningful situation. Without well-posedness, the solution might not make sense in the context of the problem being modeled. The concept of well-posedness refers to certain desirable properties that a differential equation must satisfy, which are existence, uniqueness, and continuous dependency. Regularization is an additional feature of the solution of the differential equation, such as the smoothness of the solution. Stability theory is one of the indispensable qualitative concepts of dynamical systems. We established results about the well-posedness, regularity, and Ulam-Hyers stability of solutions to conformable fractional stochastic delay differential equations. First, we discussed the results about the existence and uniqueness of solutions when the global and local Lipschitz conditions of the coefficients are satisfied. Second, we demonstrated the results about continuously depend on the fractional order $\xi$ and initial values under the global Lipschitz condition of coefficients. Thirdly, we constructed results regarding regularity and Ulam-Hyers stability, and two examples that demonstrate our results are presented. The main part of the proof made use of the truncation process, the Banach fixed point theorem, It\^{o} isometry, temporally weighted norm, Gr\"{o}nwall's inequality, and H\"{o}lder's inequality.

This study investigates the transmission dynamics of monkeypox using a fuzzy fractal fractional-order mathematical model. The model incorporates fuzzy triangular numbers into both system parameters and initial conditions to effectively represent uncertainty in real-world epidemiological data. By employing fractal-fractional derivatives, the model captures memory and hereditary effects, providing a more accurate representation of disease progression. The mathematical analysis, supported by Ulam-Hyers stability and fixed-point theory, confirms the model’s reliability. Numerical simulation reveals the effects of changing the fractional order and uncertain inputs on the susceptible, exposed, infected, and recovered populations. The results underscore the enhanced comprehension of monkeypox spread and recovery provided by the combined effects of fractional dynamics and fuzziness and provide important insights for public health policy.

ABSTRACT This paper proposes a control approach for achieving practical predefined‐time stabilization in a class of nonlinear systems influenced by matched bounded disturbances and unknown nonlinearities. Radial basis function (RBF) neural networks (NNs) are used for function approximation to tackle these uncertainties. By applying Lyapunov stability theory, we show that the proposed time‐varying control law and adaptive mechanism ensure the convergence of the system states to a region around the equilibrium within a predefined time, while all closed‐loop signals remain bounded. The effectiveness of the approach is validated through simulations on a single‐link flexible‐joint robotic manipulator.

ABSTRACT This article addresses the problem of synchronizing discrete‐time Lur'e type complex dynamic networks (CDNs) via dynamic event‐triggered control. In particular, it is considered that the control signal of each node is subject to input saturation. Using the Lyapunov Stability Theory, properties of slope‐restricted nonlinearities, and the linear matrix inequalities framework, constructive sufficient conditions are provided to ensure regional exponential synchronization of the CDN. Differently from other works in the current literature on CDNs, which require a trial‐and‐error procedure to select the event‐triggering mechanism (ETM) parameters, two systematic approaches, based on convex optimization, are presented to simultaneously synthesize (co‐design) the control law gains and the event‐generator parameters, aiming to reduce the number of events compared to a time‐triggered policy, with formal guarantees of regional synchronization with respect to a given admissible set of initial synchronization errors. Finally, a numerical example is presented to illustrate these approaches.

ABSTRACT In this paper, a formation controller for propeller‐driven car‐like robots is developed, which is subject to input amplitude, rate saturation, and steering fault‐tolerant control. First, the model of the propeller‐driven car‐like robot is established, where actuator dynamics, amplitude, and rate saturation are considered. Second, the Gauss integration function is used to approximate the input saturation. Rate saturation will be converted into command input saturation and will be achieved by an auxiliary system. Third, the formation controller is developed based on the backstepping control and fault‐tolerant control, where the adaptive robust controller and neural‐network observer are combined to deal with steering fault. According to the Lyapunov stability theory, it is proved that the propeller‐driven car‐like robot formation will be stable under the developed controller, while signals in the closed‐loop system are ultimately uniformly bounded. Finally, simulation and experiment results verify the effectiveness of the proposed formation control scheme.

Mixing and heat transfer rates are typically enhanced in high-pressure transcritical turbulent flow regimes. This is largely due to the rapid variation of thermophysical properties near the pseudo-boiling region, which can significantly amplify velocity fluctuations and promote flow destabilisation. The stability conditions are influenced by the presence of baroclinic torque, primarily driven by steep, localised density gradients across the pseudo-boiling line; an effect intensified by differentially heated wall boundaries. As a result, enstrophy levels increase compared with equivalent low-pressure systems, and flow dynamics diverge from those of classical wall-bounded turbulence. In this study the dynamic equilibrium of these instabilities is systematically analysed using linear stability theory. It is shown that under isothermal wall transcritical conditions, the nonlinear thermodynamics near the pseudo-boiling region favour destabilisation more readily than in subcritical or supercritical states; though this typically requires high-Mach-number regimes. The destabilisation is further intensified in non-isothermal wall configurations, even at low Brinkman and significantly low Mach numbers. In particular, the sensitivity of neutral curves to Brinkman number variations, along with the modal and non-modal perturbation profiles of hydrodynamic and thermodynamic modes, offer preliminary insight into the conditions driving early destabilisation. Notably, a non-isothermal set-up (where walls are held at different temperatures) is found to be a necessary condition for triggering destabilisation in low-Mach, low-Reynolds-number regimes. For the same Brinkman number, such configurations accelerate destabilisation and enhance algebraic growth compared with isothermal wall cases. As a consequence, high-pressure transcritical flows exhibit increased kinetic energy budgets, driven by elevated production rates and reduced viscous dissipation.

Multi-factor vaccination game and optimal control of a SVIS epidemic model.

New stability theory for non-autonomous fractional time-varying order derivative systems and its applications

Event-triggered ADP-based tracking controller for partially unknown nonlinear uncertain systems with input and state constraints.

Abstract Midwinter suppression, characterized by reduced baroclinic wave activity over the North Pacific storm track in January despite elevated background baroclinicity, challenges the framework of linear baroclinic instability theory. A central feature of this phenomenon is the southward shift of baroclinic waves, guided by the jet. While this shift has been identified as a key factor in midwinter suppression, the precise dynamics remain unclear, partly due to the lack of explicit terms in eddy energetics that address the impact of meridional wave shifts. In this study, we apply linear regression analysis and a simplified perturbation vorticity equation to examine vorticity amplification in baroclinic waves during their linear growth phase. Vorticity amplification is dominated by vortex stretching, a mechanism conceptualized as the concentration or dilution of planetary vorticity driven by the divergent winds associated with the wave, linking wave growth to meridional migration. Our results show that the suppression relative to November is marked by a weakening of the wave growth rate in the central-eastern Pacific and a southward shift of the growth region, with no substantial changes in wave structure. This suppression is primarily driven by the weakening of planetary vorticity due to the wave’s southward migration, which inhibits vortex stretching intensity. In contrast, the suppression relative to March is mainly attributed to the stronger westward tilt of the wave in the western Pacific in March, with no significant change in vortex stretching intensity during this period.

A sliding mode predictive control (SMPC) scheme integrated with an extreme learning machine (ELM) disturbance observer is proposed for the trajectory tracking of a flexible air-breathing hypersonic vehicle (FAHV). To streamline the controller design, the longitudinal model is decoupled into a velocity subsystem and an altitude subsystem. For the velocity subsystem, a proportional-integral sliding mode surface is designed, and the control law is derived by minimizing a cost function that weights the predicted sliding mode surface and the control input. For the altitude subsystem, a backstepping control framework is adopted, with the SMPC strategy embedded in each step. Multi-source disturbances are modeled as composite additive disturbances, and an ELM-based neural network observer is constructed for their real-time estimation and compensation, thereby enhancing system robustness. The semi-globally uniformly ultimately bounded (SGUUB) stability of the closed-loop system is rigorously proven using Lyapunov stability theory. Simulation results demonstrate the comprehensive superiority of the proposed method: it achieves reductions in Root Mean Square Error (RMSE) of 99.60% and 99.22% for velocity and altitude tracking, respectively, compared to Prescribed Performance Control with Backstepping Control (PPCBSC), and reductions of 98.48% and 97.12% relative to Terminal Sliding Mode Control (TSMC). Under parameter uncertainties, the developed ELM observer outperforms RBF-based observer and Extended State Observer (ESO) by significantly reducing tracking errors. These findings validate the high precision and strong robustness of the proposed approach.

In modern communication networks, the Transmission Control Protocol (TCP) plays a vital role in regulating end-to-end data flows. However, network parameter uncertainties and interference introduced by competing UDP flows often lead to congestion collapse, packet loss, and reduced throughput. To address these challenges, this paper proposes a novel event-triggered sliding mode control (ET-SMC) strategy for active queue management in TCP networks. This approach combines the robustness of global sliding mode control with the efficiency of event-triggered mechanisms, significantly reducing redundant control operations while maintaining system stability. Lyapunov stability theory is used to rigorously prove that all signals in the closed-loop system are bounded, effectively avoiding the Zeno phenomenon. Numerical simulations demonstrate that the ET-SMC strategy ensures queue stability, reduces control update frequency, and achieves superior performance compared to traditional PI-based and time-triggered sliding mode controllers.

Abstract Stability theory plays a central role in the analysis of the behaviour of a
solution of a real-order differential equations. In the literature, various
stability concepts were introduced from the application point of view. The
most popular ones are the Lyapunov and the Ulam-Hyers stability. Here, we
first we establish a relation between the Lyapunov and Ulam-Hyers stability
concepts for a dynamical system and prove that the concept of Ulam-Hyers
is more general than that of Lyapunov. Second, we present a brief overview
of recent developments in the Ulam-Hyers stability analysis of
fractional-order differential equations (FDEs). These equations include linear
FDEs, non-linear FDEs, delay FDEs, fractional-order boundary value problems
and impulsive FDEs.

Abstract This paper presents the modelling and fault-tolerant control of a single rotor helicopter UAV with the application of super-twisting sliding mode controller (STSMC) in the presence of actuator faults. STSMC reduces the problem of chattering found in conventional sliding mode controller (SMC) while at the same time being invariant to matched uncertainties. Unfortunately, it increases the number of controller design variables. Due to this high number of variables, the tuning process becomes too complex to perform manually. This paper proposes the design of an optimised STSMC based on the ant colony optimisation (ACO) and particle swarm optimisation (PSO) algorithms. The stability analysis of the proposed control schemes is carried out using Lyapunov stability theory. The effectiveness of these STSMC schemes for rotorcraft fault tolerance is evaluated in numerical simulations. The performance of the ACO-optimised STSMC-controlled system proved to be comparable to the PSO-optimised implementation, and both performed better than the conventional SMC. The controlled system also proved to be robust to actuator faults of up to $$30\%$$ 30 % loss of effectiveness.

Group theory is a very important concept in mathematics with many interesting theories that have been widely applied in other areas of mathematics. As one of the fundamental tools in abstract algebra, it provides a unifying language for studying symmetries, structures, and transformations, making it central to both theoretical and applied mathematics. This paper proves the orbit stability theorem based on the theory of group actions. Then, this paper introduces the application of the orbit stabilizer in other parts of mathematics and its full proof. Among these theorems, compared with other proof methods, the orbit stabilizer theorem is more concise and easier to understand. These examples show the wide application of the orbit stability theorem in mathematics, proving its practicality. Furthermore, the theorem serves as a foundation for exploring topics such as combinatorics, number theory, and geometry, where orbit-stabilizer arguments simplify otherwise complex counting and classification problems. In this way, the study highlights how group theory not only develops its own framework but also contributes essential insights to broader mathematical investigations.

This paper develops novel computational methods for studying norm-attaining functionals in infinite-dimensional Banach spaces. We present constructive approximation algorithms with explicit convergence rates, stability analysis under discretization and perturbations, and new geometric characterizations of norm attainment. Key results include: (1) efficient procedures to compute norm-attaining approximations of functionals in uniformly convex spaces, with quantitative error bounds; (2) stability theorems for finite-dimensional projections in reflexive spaces; (3) perturbation resilience estimates relating to the modulus of convexity; and (4) applications to PDE-constrained optimization and functional regression. Our approach combines techniques from functional analysis, approximation theory, and computational mathematics, yielding both theoretical insights and practical algorithms. The results significantly extend the classical Bishop-Phelps theorem by providing computable versions and quantitative estimates in various Banach space geometries.

This paper explores trajectory planning for robotic arms in 3D operational spaces. To improve the adaptation to dynamic via‐points in complex, confined environments, an enhanced dynamic movement primitives (DMP) approach is proposed and designed for dynamic planning under composite steering force field constraints. By incorporating steering attraction forces, this method enhances the generalization capability of DMP, allowing the robotic arm to navigate through dynamic via‐points flexibly without altering the start and end positions. The trajectory shape is adjusted via regression attraction forces, which helps preserve the demonstrated trajectory, reduce free‐space loss, and improve the system's adaptability to complex, dynamic environments. The convergence of the target state is rigorously proven using Lyapunov stability theory. Numerical simulations and experiments conducted with the Franka robotic arm validate the effectiveness of the proposed approach. Results show that in dynamic environments with multiple via‐points, this method produces reliable trajectories for robotic arm movements, significantly enhancing the adaptation of DMP to dynamic contexts. The planning process requires no additional learning, and the generated trajectory closely resembles the original demonstrated path. This method enables effective via‐point operations in confined spaces without requiring additional learning, while maintaining existing skills.