In this manuscript, we formulate the system of fuzzy stochastic fractional evolution equations (FSFEEs) driven by fractional Brownian motion. We find the results about the existence-uniqueness of the formulated system by using the Lipschitizian conditions. By using these conditions we have also investigated the exponential stability of the solution for the above system driven by fractional Brownian motion. Finally, the applications in financial mathematics are presented and the use of financial mathematics in the fractional Black and Scholes model is also discussed. An example is propounded to show the applicability of our results.

This study dealt with the problem of fault-tolerant control in multi-agent systems impacted also by factor stochastic impulsive disturbances together with actuator and sensor malfunctions. It is known that impulsive events can cause the agents to become unsynchronized and controlled coordination using given fault-tolerant methods is difficult. To enhance the robustness and safety of fault models, this research presented an impulsive fault-tolerant control. This controller aims to eliminate the effect of actuator faults, sensor faults, and impulsive disturbances. We also provided an overview of the sliding mode control technique, which is necessary for managing nonlinear uncertainties in impulses and faults in actuators and sensors. To describe the characteristics of the sliding mode controller, the Lyapunov function was employed to analyze its stability. The stability effect demonstrated in our paper is asymptotic stability rather than semi-global uniform ultimate boundedness because the function derivative of the Lyapunov function and its included term ensures it is negative semi-definite except for the bounded switching control terms, confirming asymptotic convergence to zero. Numerical examples were included to demonstrate the effectiveness and benefits of the proposed approach.

This paper investigates the containment of output formation for heterogeneous singular multi-agent systems (MASs) subject to time delays and stochastic impulsive disturbances. The considered agents exhibit diverse dynamics, including descriptor-type tracking and algebraic systems, reflecting heterogeneity in state-space representations. Communication constraints are modeled via bounded state and input delays, while environmental uncertainties are captured through stochastic noise and impulsive effects occurring at arbitrary time instants. A distributed control protocol is proposed to achieve output formation containment, ensuring asymptotic convergence of follower outputs to a convex combination of leader outputs. The approach effectively addresses system singularities and uncertainties by employing a Lyapunov–Krasovskii functional combined with stochastic stability analysis. Sufficient conditions for mean-square admissibility and asymptotic convergence are derived. Numerical simulations validate the effectiveness and robustness of the proposed control strategy under varying delays and impulsive conditions.

This paper presents an adaptive leader-follower formation control strategy for second-order nonlinear multi-agent systems with unknown dynamics. To handle system uncertainties, we used neural networks (NNs) to approximate and compensate for nonlinear effects. A key feature of our approach is its ability to deal with Byzantine attacks and time delays, which can disrupt coordination among agents. Unlike existing methods, our control strategy actively accounts for these challenges while ensuring stable formation tracking. Using Lyapunov stability theory, we proved that all system errors remain within a bounded range. Numerical simulations confirmed the effectiveness of our approach, showing that it successfully maintains formation control even in the presence of adversarial attacks and delays.