多智能体协调与意见动态建模中的微分不等式
Differential Inequalities in Multi-Agent Coordination and Opinion Dynamics Modeling
AI总结：
本文提出一种统一方法，通过分析微分或差分不等式来研究多智能体协调中的拉普拉斯型算法，证明其收敛性并扩展了相关研究成果。
摘要：
多智能体协调的分布式算法引起了研究界的广泛关注；最简单且研究最深入的算法是基于一般时变加权图的微分或差分方程形式的一致协议。这些图通常通过其关联的拉普拉斯矩阵进行代数表征。具有类似代数图论结构的网络算法，称为本文中的拉普拉斯型算法，在其他相关多智能体控制问题中也出现，如聚集和包含控制、目标包围、分布式优化以及社会群体中意见演变的建模。尽管它们有相似之处，但每种算法通常都使用不同的数学技术进行研究。本文提出了一种新方法，允许以统一且优雅的方式研究许多多智能体协调的拉普拉斯型算法。该方法基于分析某些微分或差分不等式，这些不等式必须被智能体的某些“输出”（例如聚集问题中的到期望集的距离）满足。虽然这些不等式可能有无数无界解，但在自然图连通性条件下，所有有界解都会收敛（甚至达成一致），从而导致相应分布式算法的收敛。在微分方程理论中，没有有界非收敛解的缺席被称为方程的二元性。本文建立了拉普拉斯型微分和差分不等式的二元性准则，并展示了这些准则能够扩展关于多智能体协调中拉普拉斯型算法以及社会群体中意见形成建模的相关研究成果。
英文摘要：
Distributed algorithms of multi-agent coordination have attracted substantial attention from the research community; the simplest and most thoroughly studied of them are consensus protocols in the form of differential or difference equations over general time-varying weighted graphs. These graphs are usually characterized algebraically by their associated Laplacian matrices. Network algorithms with similar algebraic graph theoretic structures, called being of Laplacian-type in this paper, also arise in other related multi-agent control problems, such as aggregation and containment control, target surrounding, distributed optimization and modeling of opinion evolution in social groups. In spite of their similarities, each of such algorithms has often been studied using separate mathematical techniques. In this paper, a novel approach is offered, allowing a unified and elegant way to examine many Laplacian-type algorithms for multi-agent coordination. This approach is based on the analysis of some differential or difference inequalities that have to be satisfied by the some "outputs" of the agents (e.g. the distances to the desired set in aggregation problems). Although such inequalities may have many unbounded solutions, under natural graphic connectivity conditions all their bounded solutions converge (and even reach consensus), entailing the convergence of the corresponding distributed algorithms. In the theory of differential equations the absence of bounded non-convergent solutions is referred to as the equation's dichotomy. In this paper, we establish the dichotomy criteria of Laplacian-type differential and difference inequalities and show that these criteria enable one to extend a number of recent results, concerned with Laplacian-type algorithms for multi-agent coordination and modeling opinion formation in social groups.
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