Applications at the Intersection of Hybrid Dynamical Systems and Interaction Networks

Abstract

We consider dynamical systems that exist within a larger network i.e. dynamical systems for which we have a set of local dynamics coupled together via pairwise connections. The particular coupling we have in mind relates to an object called the graph Laplacian. We further employ graph theoretic notions as a way of describing potential changes that the dynamical rules themselves may undergo. Specifically, we describe the notion of a hybrid dynamical system. We focus mainly on two different applications. First, we study protocols based on self-organizing principles, which are used to coordinate the movements of a set of multi-agents. These are meant to model a group of autonomous robotic vehicles operating independently of human control. Implementing a coordinated group of unmanned aerial vehicles, for example, is more efficient for many mission types including tracking, surveillance, and mapping. In this regard, there are numerous reasons why collective control of a group of UAVs is important. Second, inspired by the COVID-19 pandemic, we build a large scale model which accounts for coordination between regions each using travel restrictions in order to mitigate the rapid spread of disease. The model is designed via a family of compartmental SEIR models, and using the formalism of hybrid automata. There is a need for simulations of countries cooperating together, as travel restriction policies were taken by countries without global coordination. It is possible, for instance, that a strategy which appears unfavorable to a region at some point of the pandemic might be best for containing the global spread, or that only by coordinating policies among several regions can a restriction strategy be truly effective.