Martingale Convergence Theorem Research Articles

New Criteria on Asymptotic Stability of Stochastic Differential Systems via Impulsive Control

This brief is concerned with the stability issue for stochastic differential systems (SDSs). Some new criteria on asymptotic stability are established for SDSs with impulsive control based on Lyapunov stability theory, bounded difference sequences and martingale convergence theorems. The efficiency of the theory is demonstrated by two examples, i.e. a Chua’s circuit system and a numerical example, which show that the designed impulse controller can stabilize the SDSs.

Transmission-Constrained Consensus Over Random Graphs.

The exchange of information is a crucial factor in achieving consensus among agents. However, in real-world scenarios, nonideal information sharing is prevalent due to complex environmental conditions. Consider the information distortions (data) and stochastic information flow (media) during state transmission both caused by physical constraints, a novel model of transmission-constrained consensus over random networks is proposed in this work. The transmission constraints are represented by heterogeneous functions that reflect the impact of environmental interference in multiagent systems or social networks. A directed random graph is applied to model the stochastic information flow where every edge is connected probabilistically. Using stochastic stability theory and the martingale convergence theorem, it is demonstrated that the agent states will converge to a consensus value with probability 1, despite information distortions and randomness in information flow. Numerical simulations are presented to validate the effectiveness of the proposed model.

Martingale convergence theorems for tensor splines

Martingale convergence theorems for tensor splines

A proof and an application of the continuous parameter martingale convergence theorem

A proof and an application of the continuous parameter martingale convergence theorem

The balanced split step theta approximations of stochastic neutral Hopfield neural networks with time delay and Poisson jumps

The balanced numerical approximations of the stochastic neutral Hopfield neural networks (SNHNN) with time delay and Poisson jumps are examined to ascertain the nature of the mathematical model. The numerical approximations of the balanced split-step theta methods for the SNHNN with time delay and Poisson jumps are taken into consideration primarily because they maintain almost surely (a.s.) exponential stability property of numerical methods and produce negligible mean square error when compared to other approaches. Furthermore, in the recent development of numerical approximations for SNHNN with time delay, we note that balanced split-step theta-approximations are a more stable scheme. We showed that the numerical approximations of balanced split-step theta methods of SNHNN with time delay and Poisson jumps have strong convergence order 1/2 and are numerically almost exponentially stable by applying some theoretical significance criteria.Moreover, our main research tools are Lipschitz conditions, linear growth conditions, and the discrete semi martingale convergence theorem. Through numerical experiments, we try to demonstrate the theoretical results obtained in this paper. Finally, we got the confirmation about the theoretical results of the split-step theta-approximations of SNHNN with time delay and Poisson jumps via particular numerical example.

Resilient Vector Consensus Over Random Dynamic Networks Under Mobile Malicious Attacks

AbstractThis paper investigates the problem of resilient vector consensus for a group of dynamic agents against mobile malicious attacks. As real networks often operate under random environment and noises, we approach this problem by considering general random dynamic networks with weighted directed topologies. We propose three types of mobile attack models, which differ in the timing of moving of attackers and the capability of detecting such moving. By employing distributed discrete-time algorithms, the Lyapunov theory and martingale convergence theorem, resilient vector consensus is shown to be reached for all three models when the underlying network satisfies certain stochastic robustness conditions.

ON THE IDEAL CONVERGENCE OF MARTINGALES

This paper deals with the ideal convergence theorem of martingales of ideal Bochner integrablefunctions.First, a characterization of the properties of ideal convergent martingales is obtained.In this papermartingales of ideal Bochner integrable functions with values in a Banach space are treated.

Stability Analysis for a Class of Stochastic Differential Equations with Impulses

This paper is concerned with the problem of asymptotic stability for a class of stochastic differential equations with impulsive effects. A sufficient criterion on asymptotic stability is derived for such impulsive stochastic differential equations via Lyapunov stability theory, bounded difference condition and martingale convergence theorem. The results show that the impulses can facilitate the stability of the stochastic differential equations when the original system is not stable. Finally, the feasibility of our results is confirmed by two numerical examples and their simulations.

Consensus Disturbance Rejection of Linear Discrete-Time Multi-Agent Systems With Communication Noises

This paper studies consensus disturbance rejection problem of discrete-time multi-agent systems with communication noises. The involved agents are modeled by linear dynamics with disturbance, and are assumed to receive information corrupted by additive communication noises receiving from their neighbours. The control aim, not only is to guarantee the average state of the agents that is subject to deterministic disturbance, but also is to exclude the communication noises. In order to achieve this control aim, some disturbance observers are designed under two kinds of stochastic-approximation type control gains. By utilizing the martingale convergence theorem and some mathematical techniques, two sufficient conditions are obtained to ensure consensus and disturbance rejection in mean square. Finally, a numerical example is given to show the applicability of considered methods.

On a quantum martingale convergence theorem

It is well known in quantum information theory that a positive operator-valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalized POVM, namely, a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator-valued function which is measurable with respect to a quantum probability. In this work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue-dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits nonclassical behavior; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.

Resilient tracking consensus over dynamic random graphs: A linear system approach

Cooperative coordination in multi-agent systems has been a topic of interest in networked control theory in recent years. In contrast to cooperative agents, Byzantine agents in a network are capable to manipulate their data arbitrarily and send bad messages to neighbors, causing serious network security issues. This paper is concerned with resilient tracking consensus over a time-varying random directed graph, which consists of cooperative agents, Byzantine agents and a single leader. The objective of resilient tracking consensus is the convergence of cooperative agents to the leader in the presence of those deleterious Byzantine agents. We assume that the number and identity of the Byzantine agents are not known to cooperative agents, and the communication edges in the graph are dynamically randomly evolving. Based upon linear system analysis and a martingale convergence theorem, we design a linear discrete-time protocol to ensure tracking consensus almost surely in a purely distributed manner. Some numerical examples are provided to verify our theoretical results.

Distributed Order Estimation of ARX Model under Cooperative Excitation Condition

In this paper, we consider the distributed estimation problem of a linear stochastic system described by an autoregressive model with exogenous inputs when both the system orders and parameters are unknown. We design distributed algorithms to estimate the unknown orders and parameters by combining the proposed local information criterion with the distributed least squares method. The simultaneous estimation for both the system orders and parameters brings challenges for the theoretical analysis. Some analysis techniques, such as double array martingale limit theory, stochastic Lyapunov functions, and martingale convergence theorems are employed. For the case where the upper bounds of the true orders are available, we introduce a cooperative excitation condition, under which the strong consistency of the estimation for the orders and parameters is established. Moreover, for the case where the upper bounds of true orders are unknown, a similar distributed algorithm is proposed to estimate both the orders and parameters, and the corresponding convergence analysis for the proposed algorithm is provided. We remark that our results are obtained without relying on the independency or stationarity assumptions of regression vectors, and the cooperative excitation conditions can show that all sensors can cooperate to fulfill the estimation task even though any individual sensor cannot.

Median-Based Resilient Consensus Over Time-Varying Random Networks

This brief investigates the resilient consensus control for multiagent systems over a time-varying directed random network. We propose a median-based consensus strategy, which is purely distributed and, as opposed to the Weighted-Mean-Subsequence-Reduced approaches in the existing literature, shared estimate regarding the number of malicious agents in the neighborhood of each cooperative agent is not required. This offers more applicability and flexibility as seeking a shared estimate of surrounding threats is often difficult in practice. In addition to malicious agents, random availability of communication edges is accommodated in the random network framework. Sufficient conditions are derived for reaching almost sure consensus by using a martingale convergence theorem. Finally, the theoretical findings are illustrated by numerical simulations.

Martingale convergence theorem for a conditional intuitionistic fuzzy mean value

The aim of this contribution is to show a representation of a conditional intuitionistic fuzzy mean value of intuitionistic fuzzy observables by a conditional mean value of random variables. We formulate a martingale convergence theorem for a conditional intuitionistic fuzzy mean value, too.

Stochastic bipartite consensus with measurement noises and antagonistic information

This paper is dedicated to the stochastic bipartite consensus issue of discrete-time multi-agent systems subject to additive/multiplicative noise over antagonistic network, where a stochastic approximation time-varying gain is utilized for noise attenuation. The antagonistic information is characterized by a signed graph. We first show that the semi-decomposition approach, combining with Martingale convergence theorem, suffices to assure the bipartite consensus of the agents that are disturbed by additive noise. For multiplicative noise, we turn to the tool from Lyapunov-based technique to guarantee the boundedness of agents’ states. Based on it, the bipartite consensus with multiplicative noise can be achieved. It is found that the constant stochastic approximation control gain is inapplicable for the bipartite consensus with multiplicative noise. Moreover, the convergence rate of stochastic MASs with communication noise and antagonistic exchange is explicitly characterized, which has a close relationship with the stochastic approximation gain. Finally, we verify the obtained theoretical results via a numerical example.

Martingale Convergence Theorem for the Conditional Intuitionistic Fuzzy Probability

For the first time, the concept of conditional probability on intuitionistic fuzzy sets was introduced by K. Lendelová. She defined the conditional intuitionistic fuzzy probability using a separating intuitionistic fuzzy probability. Later in 2009, V. Valenčáková generalized this result and defined the conditional probability for the MV-algebra of inuitionistic fuzzy sets using the state and probability on this MV-algebra. She also proved the properties of conditional intuitionistic fuzzy probability on this MV-algebra. B. Riečan formulated the notion of conditional probability for intuitionistic fuzzy sets using an intuitionistic fuzzy state. We use this definition in our paper. Since the convergence theorems play an important role in classical theory of probability and statistics, we study the martingale convergence theorem for the conditional intuitionistic fuzzy probability. The aim of this contribution is to formulate a version of the martingale convergence theorem for a conditional intuitionistic fuzzy probability induced by an intuitionistic fuzzy state m. We work in the family of intuitionistic fuzzy sets introduced by K. T. Atanassov as an extension of fuzzy sets introduced by L. Zadeh. We proved the properties of the conditional intuitionistic fuzzy probability.

Conditional intuitionistic fuzzy probability and martingale convergence theorem using IF-probability

The aim of this paper is to formulate the conditional intuitionistic fuzzy probability and a version of martingale convergence theorem with respect an intuitionistic fuzzy probability. Since the intuitionistic fuzzy probability can be decomposed to two intuitionistic fuzzy states, we can use the results holding for intuitionistic fuzzy states.

Almost everywhere convergence of spline sequences

We prove the analogue of the Martingale Convergence Theorem for polynomial spline sequences. Given a natural number k and a sequence (ti) of knots in [0, 1] with multiplicity ≤ k − 1, we let Pn be the orthogonal projection onto the space of spline polynomials in [0, 1] of degree k − 1 corresponding to the grid $$\left( {{t_i}} \right)_{i = 1}^n$$ . Let X be a Banach space with the Radon—Nikodým property. Let (gn) be a bounded sequence in the Bochner—Lebesgue space $$L_X^1$$ [0, 1] satisfying $${g_n} = {P_n}\left( {{g_{n + 1}}} \right),\,\,\,\,\,n \in\mathbb{N} .$$ We prove the existence of limn→∞gn (t) in X for almost every t ∈ [0, 1]. Already in the scalar valued case X = ℝ the result is new.

Almost Sure Stability of Nonlinear Systems Under Random and Impulsive Sequential Attacks

This article is concerned with the stability problem for a class of Lipschitz-type nonlinear systems in networked environments, which are suffered from random and impulsive deception attacks. The attack is modeled as a randomly destabilizing impulsive sequence, whose impulsive instants and impulsive gains are both random with only the expectations available. Almost sure stability is ensured based on Doob's Martingale Convergence Theorem. Sufficient conditions are derived for the solution of the nonlinear system to be almost surely stable. An example is given to verify the effectiveness of the theoretical results. It is shown that the random attack will be able to destroy the stability, therefore, a large feedback gain may be necessary.

Parameter estimation for dual-rate sampled Hammerstein systems with dead-zone nonlinearity

The identification of nonlinear systems with multiple sampled rates is a difficult task. The motivation of our paper is to study the parameter estimation problem of Hammerstein systems with dead-zone characteristics by using the dual-rate sampled data. Firstly, the auxiliary model identification principle is used to estimate the unmeasurable variables, and the recursive estimation algorithm is proposed to identify the parameters of the static nonlinear model with the dead-zone function and the parameters of the dynamic linear system model. Then, the convergence of the proposed identification algorithm is analyzed by using the martingale convergence theorem. It is proved theoretically that the estimated parameters can converge to the real values under the condition of continuous excitation. Finally, the validity of the proposed algorithm is proved by the identification of the dual-rate sampled nonlinear systems.