A Candidate Lyapunov Function

Based on an assignment graph $G$ , we define a candidate Lyapunov function $V: \mathbb{R}^{2n} \to \mathbb{R}$ as

$$V(x) = \underset{e_{i,j} \in E(G)}{\sum} \ell_{i,j},$$ (6)

This definition of $V$ is insensitive to edge directions in $G$ . Unlike more conventional quadratic Lyapunov functions, this specific $V$ takes a form linear in $\ell_{i,j}$ , which means that the corresponding time derivative is not linear in $\ell_{i,j}$ or $p_i$ . The reason for the choice of the Lyapunov function will become clear in the next subsection. When a merge happens, we assume that the distance between the two merged agents is frozen, which is achievable by synchronizing their control. This gives us a continuous $V$ . Alternatively, we could also let the two merged agents have zero distance, which would induce a jump in $V$ . Since a system with $n$ agents only has up to $(n-1)$ merges, these discontinuities would not affect the overall rendezvous behavior of the system.

The function $V$ defined in (6) is clearly a graph-compatible Lyapunov function. By Lemma 2, it is rendezvous positive definite if and only if $G$ is connected. We may then study the behavior of this Lyapunov function over single-target assignment graphs to derive sufficent conditions for rendezvous of the system. For moving agents, $V$ can be considered as a function of time; its time derivative is then

$$\dot{V} = \underset{e_{i,j} \in E(G)}{\sum}\dot{\ell}_{i,j}.$$ (7)