This research presents the formulation of an anti-disturbance secure proportional integral retarded tracking control for singularly perturbed semi-Markov jump nonlinear systems in the presence of hybrid attacks, disturbances and parametric uncertainties. Typically, a Luenberger observer-oriented parallel equivalent input disturbance estimator is crafted for precise evaluations of system states and disturbances that impact plant's behaviour. Furthermore, the assessments are subsequently deployed within the configured proportional integral retarded tracking control algorithm consisting of three loops and an anti-disturbance tracking control law is configured to procure adequate tracking objectives. Along with its design, the constructed observer is equipped with supplementary security features to defend the risks posed by hybrid cyber attacks. Collectively, an observer-based anti-disturbance proportional integral tracking control with state and disturbance assessments is crafted to manifest consistent tracking goals for the considered singularly perturbed semi-Markov jump systems. Thereafter, by leveraging appropriate Lyapunov–Krasovskii functionals, the verification of the system's stochastic stability is systematically conditioned as linear-matrix-inequality. Further, to strengthen the consistency of procured results, graphical illustrations are presented in the form of numerical simulations as supporting evidence.

This paper concerns the issue of event-triggered control (ETC) for a class of continuous-time Markov jump system subject to packet losses and deception attacks. In view of inevitable imperfections of communication, triggered data are able to be lost and suffer from deception attacks over networks, which might lead to Zeno phenomenon. To tackle this problem, a strict positive lower bound is embedded in event-triggered scheme (ETS) in advance and the triggering inequality under packet losses and deception attacks is re-estimated. Considering that the probability of deception attacks may depend on the mode information of subsystems, hidden Markov model (HMM) is employed. Then via Lyapunov function method, sufficient conditions involving the number of consecutive packet losses and conditional probability of HMM are provided to ensure the stochastic stability of Markov jump systems. Finally, the correctness and effectiveness of the developed control method is demonstrated by means of a numerical example.

ABSTRACT This paper investigates strategic dynamics under the behavioral rule of pairwise interact and imitate (PII), which requires minimal information and emphasizes outperforming opponents in pairwise interactions. We characterize PII using weak tournament graphs and, for a broad class of dynamics, establish a one‐shot stability result for stochastic stability. Applications include Cournot competition, strategic complements and substitutes, externalities, Nash demand games, and status‐seeking contests. The analysis highlights the competing roles of spite effects and perturbations in favoring relative success versus efficiency.

This article investigates the stochastic asymptotic stability, boundedness, and square integrability of solutions to a class of second-order nonlinear stochastic integro-differential equations with multiple variable delays. The analysis is conducted through the construction of an appropriate Lyapunov-Krasovskii functional (L-KF), tailored to handle the combined effects of stochastic perturbations, time-varying delays, and integro-differential memory terms. Unlike many existing studies, our framework accommodates all these complexities simultaneously, thereby generalizing and extending recent contributions in the field while relaxing several restrictive assumptions. To validate the theoretical results and illustrate their practical applicability, numerical simulations are provided.

Deep learning systems in digital pathology are widely regarded as opaque, limiting clinical trust and interpretability. We present a framework for empirically characterizing training-time learning dynamics in neural networks by directly measuring activation structure, weight evolution, and spectral organization during optimization. Using TCGA BRCA whole-slide images with replication-timing derived methylation proxies as regression targets, we trained a Vision Transformer and tracked its intra-epoch behavior across 20 epochs.We observed reproducible structural signatures during training. Correlated groups of neurons formed stable activation modules whose modularity increased as training progressed, accompanied by a reduction in representation entropy of up to 60\\%. Weight trajectories exhibited bounded diffusion with progressively reduced variance, consistent with a damped stochastic process, and converged toward a stable stationary regime in later epochs. In image space, model attention systematically shifted from collagen-rich stromal regions in early epochs to basophilic, proliferative nuclear regions in later epochs, aligning with known histologic correlates of replicative stress.These findings demonstrate that neural networks develop predictable, quantifiable internal structure during training that can be directly visualized and measured. Framing learning dynamics in terms of entropy, modular organization, and stochastic stabilization provides a practical, mechanistic lens for interpreting how pathology AI models acquire biologically meaningful representations.

Stochastic dynamic stability assessment with high penetration of uncertain sources: Uncertainty modeling, solutions and applications

Stochastic stability of positive Markov jump subhomogeneous and homogeneous nonlinear systems

Abstract A system of stochastic differential equations is proposed as a model to describe the diffusion of malware in a wireless sensor network. The system is obtained from a SEIR model by perturbing the rate at which susceptible sensors become exposed using a Gaussian white noise. Then, the stochastic stability of the model is analysed to obtain a sufficient condition for the asymptotic stochastic stability of the disease-free equilibrium.

This paper investigates the stability properties of stochastic functional differential equations driven by fractional Brownian motion (FBM), a natural extension of classical Brownian motion that incorporates memory effects through the Hurst parameter. By constructing appropriate Lyapunov functionals, we derive sufficient conditions for global exponential mean square stability and asymptotic stochastic stability. The analysis is applied to a delayed stochastic fractional Black–Scholes model, where explicit criteria for both stability types are established. Numerical simulations are presented to validate the theoretical results and illustrate the influence of the Hurst index on the stability behavior of the system.

The parameter perturbation makes it difficult to analyze the dynamic mechanism of the brake chatter with the help of an ideal deterministic model. Hence, a stochastic dynamic model of brake chatter with the uncertainty of the torsional stiffness of the brake disc is established, the Stribeck model is used to describe the friction characteristic between the brake pad and brake disc. Then, the Itô stochastic differential equation is solved by means of the stochastic averaging method, the boundary type of the one-dimensional energy process is identified to discuss the stochastic stability of the brake system. Furthermore, the Fokker Planck Kolmogorov equation is derived, the dynamic bifurcation and phenomenological bifurcation are analytically proven, the influences of parameters, such as noise intensity, brake pressure, and friction coefficient, on bifurcation characteristics are revealed. The results show that the noise intensity, brake pressure, and friction coefficient difference need to be reduced to improve the system stability. Finally, the probability density function of the brake system is used to validate the analytical analysis. The relevant results provide a theoretical basis for better suppressing the brake chatter.

Model-free robust tracking control of LTI systems with Markov jump target: A data informativity approach.

Parametric stochastic vibration and stability analysis of axially moving shape memory alloy strip plates

This article addresses the time-varying distributed optimization problem (DOP) for networked multiagent systems (NMASs) operating over directed graphs, considering the impact of edge-based additive measurement noise (EBAMN). First, a finite-time stochastic stability framework is established to demonstrate the global stochastic practical finite-time attraction of the origin, enabling robust control design for stochastic nonlinear systems. The proposed method achieves faster convergence rates and provides bounded finite convergence time estimates, outperforming asymptotic methods. Second, a novel distributed optimization algorithm (DOA) is introduced, incorporating consensus-gain function, state-dependent optimization gains, and integral information of the gradient of local objective functions. Using the It $\mathrm {\hat {o}}$ lemma and Lyapunov theory, the continuous-time DOA guarantees the $p$ th moment convergence for all agents, ensures practical finite-time consensus in probability, and drives that states of NMASs converge to the time-varying optimal solution, even in the presence of EBAMN interferences. Furthermore, a new adaptive dynamic event-triggered mechanism (ETM) integrated with the DOA is proposed. This mechanism significantly enhances communication efficiency and reduces resource consumption throughout the process of tracking the optimal solution while preventing Zeno behavior. Finally, numerical simulations in multiuncrewed aerial vehicle (UAV) target tracking validate the effectiveness of the robust continuous-time DOA against random EBAMN.

This article addresses the output feedback control problem for a specific class of discrete-time fuzzy singularly perturbed systems subjected to nonuniform sampling and a round-robin protocol. An innovative method for modeling nonuniform sampling periods through nonhomogeneous sojourn probabilities is proposed, offering a more intuitive and adaptable framework for system design and analysis. The round-robin protocol is applied to nonuniformly sampled outputs, optimizing information transmission efficiency and enhancing overall system performance. To tackle potential limitations in state data acquisition, a token-dependent static output feedback controller is developed that addresses the complexities introduced by nonperiodic sampling and asynchronous premise variables. Sufficient conditions are derived to ensure stochastic stability of the closed-loop system. Finally, two simulation examples are presented to validate and demonstrate effectiveness of the theoretical approach.

This article addresses the lateral dynamics control problem for autonomous vehicle systems under randomly perturbed sampling (RPS) periods and the FlexRay communication protocol. To capture vehicle nonlinearities under variable-velocity conditions, a T-S fuzzy model is constructed using longitudinal velocity as the premise variable. The random sampling behavior caused by hardware aging and environmental disturbances is modeled as a Markovian process. Then, measured outputs are transmitted under the FlexRay protocol (FRP) that integrates both time-driven (static) and event-driven (dynamic) scheduling characteristics. By fully analyzing the situation of static and dynamic scheduling, a unified compensation strategy is employed to build a new switching output model reflecting the impact of the FRP on the measured outputs. Based on this output model, a sampling-mode-dependent fuzzy controller is designed to handle random sampling and hybrid scheduling issues, which results in a membership asynchronous phenomenon between the autonomous vehicle model and controller. By using the asynchronous constraint technique, sufficient conditions with low conservatism are derived to guarantee stochastic stability and $H_{\infty }$ performance of the closed-loop system. Furthermore, a comprehensive optimization problem (OP) is established, and a corresponding genetic algorithm (GA) is presented to provide a solution-solving scheme. Simulation results confirm the effectiveness and superiority of the proposed control strategy under complex communication environments.

This article investigates the mean square exponential stability for dynamic memristor-neutral stochastic cellular neural networks with time-varying delays (DM-NSDCNNs). Unlike general neural networks (NNs) analyzed in the voltage-current domain, DM-NSDCNNs are studied in the flux-charge domain, offering a significant advantage: all current, voltage, and power consumption vanish when the system reaches a steady state. In particular, dynamic memristor store the results of computation. To better utilize these properties, two distinct stochastic stability analysis techniques are considered, depending on the memristor's constitutive relations. For piecewise linear constitutive relation, the stability criteria are obtained by a novel approach based onthe comparison principle and reductio ad absurdum. Moreover, the stability criteria for cubic nonlinear constitutive relation are established via stochastic analysis employing Lyapunov functional techniques. Finally, several numerical examples with different constitutive relations of DM-NSDCNNs are provided to verify the effectiveness and potential of the proposed results.

Abstract In this paper, we study the existence and the uniqueness of optional solutions of stochastic differential equations with respect to optional semimartingales by using a successive approximation method under a non-Lipschitz condition. The stability of solutions to non-Lipschitz SDEs is also considered, and the stochastic stability is obtained in the sense of mean square.

Abstract The paper presents the dynamical analysis of an avian influenza infection model in birds population with half-saturated incidence. The positivity of solutions is proved by the occurrence of a global positive solution. Also, we analyze the extinction scenario of infection under certain parametric restriction. Moreover, we define the reproduction ratio R 0 S R_{0}^{S} to examine the stochastic stability through the stationary distribution. In order to figure out when infection appears in the birds population and when it stops the spread, we derive the condition on R 0 S R_{0}^{S} . Furthermore, we generate some graphical simulations to support the theoretical results based on the value of reproduction ratio.

This paper presents a comprehensive analysis of non-fragile projective lag synchronization (PLS) in semi-Markov jumping quaternion-valued neural networks (SMQVNNs) with complex delay structures, encompassing leakage delay, proportional delays, and distributed delays. To address system uncertainties, a non-fragile controller is incorporated, while the benefits of sliding mode control (SMC) are leveraged to enhance robustness. The integration of non-fragile control and SMC enables the system to effectively mitigate uncertainties and disturbances. The sliding surface is formulated using integral terms to ascertain the existence of sliding motion, and the direct Lyapunov technique is used to prove the SMC's effectiveness in driving the error system towards the target sliding surface. Furthermore, the error system's stochastic asymptotic stability is examined, and a stability condition is formulated. Numerical simulations, along with comparative results, confirm that the proposed method achieves faster and more robust synchronization than existing control schemes.

ABSTRACT This article addresses the fault estimation problem for a class of discrete‐time Markovian jump systems affected by disturbances. A novel modified observer is proposed to simultaneously estimate the system state and fault, considering both constant and time‐varying faults. By constructing a stochastic Lyapunov functional, a sufficient and solvable condition in terms of linear matrix inequalities is established to ensure the stochastic stability of the overall error dynamics system. An performance index is employed to mitigate the effects of unknown disturbances and fault variations. Subsequently, the fault estimation strategy is utilized to address the fault detection problem. Finally, the proposed approach is validated through two practical examples: a boost converter and a mass‐spring‐damper system, illustrating the applicability and feasibility of the theoretical results.