Bounded-confidence models in social dynamics describe multi-agent systems, where each individual interacts only locally with others. Several models are written as systems of ordinary differential equations (ODEs) with discontinuous right-hand side: this is a direct consequence of restricting interactions to a bounded region with non-vanishing strength at the boundary. Various works in the literature analyzed properties of solutions, such as barycenter invariance and clustering. On the other side, the problem of giving a precise definition of solution, from an analytical point of view, was often overlooked. However, a rich literature proposing different concepts of solution to discontinuous differential equations is available. Using several concepts of solution, we show how existence is granted under general assumptions, while uniqueness may fail even in dimension one, but holds for almost every initial conditions. Consequently, various properties of solutions depend on the useddefinition and initial conditions.

中文翻译：

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有限置信模型的广义解

社会动力学中的有限置信模型描述了多智能体系统，其中每个人仅在本地与他人交互。几个模型被写成右手边不连续的常微分方程 (ODE) 系统：这是将相互作用限制在边界处具有非消失强度的有界区域的直接结果。文献中的各种作品分析了解决方案的属性，例如重心不变性和聚类。另一方面，从分析的角度来看，给出解决方案的精确定义的问题经常被忽视。但是，有大量文献提出了不连续微分方程的不同解概念。使用几个解决方案的概念，我们展示了在一般假设下如何授予存在，虽然即使在第一维中唯一性也可能失败，但几乎适用于每个初始条件。因此，解的各种性质取决于使用的定义和初始条件。