This paper is concerned with the stabilization problem for nonlinear stochastic delay systems with Markovian switching by feedback control based on discrete-time state and mode observations. By constructing an efficient Lyapunov functionals, we establish the sufficient stabilization criteria not only in the sense of exponential stability (both the mean-square stability and the almost sure stability) but also in other sense – that of $H_{\mathrm{\infty }}$ stability and asymptotic stability. Meanwhile, the upper bound on the duration τ between two consecutive state and mode observations is obtained. Numerical examples are provided to demonstrate the effectiveness of our theoretical results.

This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both $s=t$ and $s=0$. We propose a Galerkin approximate scheme to solve the equation numerically, and we obtain the strong convergence rate for the Galerkin method in the mean square sense. The rate is $min\{2-2({\alpha }_1+{\beta }_1),1-2({\alpha }_2+{\beta }_2)\}$ (where ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2$ are positive numbers satisfying $0<{\alpha }_1+{\beta }_1<1$, $0<{\alpha }_2+{\beta }_2<\frac{1}{2}$), which improves the results of some numerical schemes for the stochastic Volterra integral equations with regular or weakly singular kernels. Moreover, numerical examples are given to support the theoretical result and explain the priority of the Galerkin method.

This paper investigates the robust stabilization problem for a class of neutral uncertain stochastic systems with Markovian switching and time-varying delays. The objective is to design a feedback controller based on discrete-time state observations and continuously available mode information such that the closed-loop system achieves robust exponential stability in the mean-square sense against all admissible uncertainties. By employing the Razumikhin technique, we derive sufficient conditions for mean-square exponential stability of the controlled system without imposing any differentiability assumption on the time-varying delay—only boundedness is required. This significantly relaxes the commonly used conditions in the existing literature, where the delay is required to be differentiable with a derivative bounded by a constant strictly less than one, or to satisfy a measure-type condition for non-differentiable delays. On this basis, the desired discrete-time feedback controller is explicitly constructed via linear matrix inequalities (LMIs), and an explicit estimate of the corresponding Lyapunov exponent is provided, which is rarely addressed in existing discrete-time feedback control results for neutral systems. Finally, numerical examples illustrate the effectiveness of the controller and demonstrate the applicability of the proposed criterion to a sampling-holding delay that violates commonly used differentiability and measure-type delay conditions.

This paper investigates how opinion dynamics with memory effects and random communication uncertainties achieve consensus in an inhomogeneous Hegselmann–Krause (H-K) model. The Caputo fractional derivative is introduced to describe the influence of historical opinions, while multiplicative noise captures random disturbances in communication. We establish sufficient conditions under which all followers almost surely converge to the leader’s opinion. Moreover, we characterize the algebraic (power-law) convergence behavior determined by the fractional order. The analysis is based on fractional Lyapunov techniques and fractional Grönwall estimates, which enable the treatment of the nonlinear stochastic system with leadership. The results reveal how memory effects, leadership strength and stochastic perturbations jointly influence consensus formation. Numerical simulations are provided to verify the theoretical results and illustrate the effects of the model parameters on the convergence dynamics.