Herdability of Laplacian Game Control Systems

Herdability is a qualitative property indicating that the state trajectory of a controlled system can be driven to exceed any specified non-negative threshold. A representative application of this concept is in political elections, where a candidate aims to elevate their approval rating beyond a certain positive percentage to secure victory. In this context, gaming and strategic interactions emerge as critical factors influencing herdability. Motivated by these considerations, this study investigates the herdability of Laplacian game control (LGC) systems through the lens of graph topology. Herdability is generally easier to achieve than controllability, as it pertains to the reachability of a specific subset of the state space, whereas controllability involves reachability of the entire state space. This paper first investigates the controllability of LGC systems and derives the necessary conditions. Subsequently, the distinctions between controllability and herdability of LGC systems are analyzed under identical graph-theoretic conditions. Finally, the concept of herd graphs is introduced as a useful analytical tool for establishing graph-theoretic conditions for herdability, thereby facilitating the assessment of herdability based on the system's underlying topology. The exposition of the notions and results is supported with two examples.