Xiaofeng Zong

Xiaofeng Zong, Tao Li, Ji-Feng Zhang

This work is concerned with stochastic consensus conditions of multi-agent systems with both time-delays and measurement noises. For the case of additive noises, we develop some necessary conditions and sufficient conditions for stochastic weak consensus by estimating the differential resolvent function for delay equations. By the martingale convergence theorem, we obtain necessary conditions and sufficient conditions for stochastic strong consensus. For the case of multiplicative noises, we consider two kinds of time-delays, appeared in the measurement term and the noise term, respectively. We first show that stochastic weak consensus with the exponential convergence rate implies stochastic strong consensus. Then by constructing degenerate Lyapunov functional, we find the sufficient consensus conditions and show that stochastic consensus can be achieved by carefully choosing the control gain according to the noise intensities and the time-delay in the measurement term.

Xiaofeng Zong, Fuke Wu, Guiping Xu

This paper examines convergence and stability of the two classes of theta-Milstein schemes for stochastic differential equations (SDEs) with non-global Lipschitz continuous coefficients: the split-step theta-Milstein (SSTM) scheme and the stochastic theta-Milstein (STM) scheme. For θ\in[1/2,1], this paper concludes that the two classes of theta-Milstein schemes converge strongly to the exact solution with the order 1. For θ\in [0,1/2], under the additional linear growth condition for the drift coefficient, these two classes of the theta-Milstein schemes are also strongly convergent with the standard order. This paper also investigates exponential mean-square stability of these two classes of the theta-Milstein schemes. For θ\in(1/2, 1], these two theta-Milstein schemes can share the exponential mean-square stability of the exact solution. For θ\in[0, 1/2], similar to the convergence, under the additional linear growth condition, these two theta-Milstein schemes can also reproduce the exponential mean-square stability of the exact solution.

Xuping Hou, Xiaofeng Zong, Yong He

This work aims to address the design of fully distributed control protocols for stochastic consensus, and, for the first time, establishes the existence and uniqueness of solutions for the path-dependent and highly nonlinear closed-loop systems under both undirected and directed topologies, bridging a critical gap in the literature. For the case of directed graphs, a unified fully distributed control protocol is designed for the first time to guarantee mean square and almost sure consensus of stochastic multi-agent systems under directed graphs. Moreover, an enhanced fully distributed protocol with additional tunable parameters designed for undirected graphs is proposed, which guarantees stochastic consensus while achieving superior convergence speed. Additionally, our work provides explicit exponential estimates for the corresponding convergence rates of stochastic consensus, elucidating the relationship between the exponential convergence rate and the system parameters. Simulations validate the theoretical results.

Yan Chen, Tao Li, Xiaofeng Zong

We study the distributed optimization problem over a graphon with a continuum of nodes, which is regarded as the limit of the distributed networked optimization as the number of nodes goes to infinity. Each node has a private local cost function. The global cost function, which all nodes cooperatively minimize, is the integral of the local cost functions on the node set. We propose stochastic gradient descent and gradient tracking algorithms over the graphon. We establish a general lemma for the upper bound estimation related to a class of time-varying differential inequalities with negative linear terms, based upon which, we prove that for both kinds of algorithms, the second moments of the nodes' states are uniformly bounded. Especially, for the stochastic gradient tracking algorithm, we transform the convergence analysis into the asymptotic property of coupled nonlinear differential inequalities with time-varying coefficients and develop a decoupling method. For both kinds of algorithms, we show that by choosing the time-varying algorithm gains properly, all nodes' states achieve $\mathcal{L}^{\infty}$-consensus for a connected graphon. Furthermore, if the local cost functions are strongly convex, then all nodes' states converge to the minimizer of the global cost function and the auxiliary states in the stochastic gradient tracking algorithm converge to the gradient value of the global cost function at the minimizer uniformly in mean square.