Guaranteed Rendezvous of Agents with Bounded, Varying Speeds

(33) |

The structures, , , and the hyperplane from Lemma 7 remain the same. For the slice by the hyperplane, applying the method of Lagrange multipliers to as a function of 's with constraint yields that for all ,

Once again, holding 's fixed, for to take maximum on the slice, for all , must take the same value and therefore, must be positive if we keep . Let us assume that we pick some , then

(35) |

By the pigeonhole principle, for at least one , . Therefore,

To make , we need , which is true if

(37) |

One way to satisfy this is to make sure that for each ,

or equivalently,

The right side of (39) achieves the global minimum when , which gives us (32) as a sufficient condition for on the slice. On the slice, we have that , which is less than the last expression in (36); therefore, (32) also works for the slice. The finite time guarantee follows the argument from Corollary 9.

Moving to the intree case, when agents have different speeds, no equivalent of Lemma 11 can be stated since can no longer be guaranteed. A simple example is illustrated in Fig. 5. Agent is at the root and does not move. Suppose agent moves very fast and agents ( in the figure) following barely move. Also assume that all agents are almost colinear. It is straightforward to see that, after a short period of time (the second drawing), the sum of the length of all edges, or , increases. This suggests that must be positive at some point. However, such a system will still rendezvous. Supposing that the stationary agent is , at least one agent, say , is assigned to . Thus, whenever . Hence, agent will merge into agent in finite time, and all other agents will eventually follow. We have proved:

Since it is not possible to guarantee at all times for the bounded speed case, a result like Theorem 12 is out of the question. However, since Theorem 13 and Lemma 14 parallel Theorem 4 and Lemma 11, the argument giving us sequential rendezvous in the identical agents case continues to hold: