Bounds on Sweep-Covers by Functional Compositions of Ordered Integer Partitions and Raney Numbers

A sweep-cover is a vertex separator in trees that covers all the nodes by some ancestor-descendent relationship. This work provides an algorithm for finding all sweep-covers of a given size in any tree. The algorithm's complexity is proven on a class of infinite $\Delta$-ary trees with constant path lengths between the $\Delta$-star internal nodes. I prove the enumeration expression on these infinite trees is a recurrence relation of functional compositions on ordered integer partitions. The upper bound on the enumeration is analyzed with respect to the size of sweep cover $n$, maximum out-degree $\Delta$ of the tree, and path length $\gamma$, $O(n^n)$, $O(\Delta^c c^\Delta)$, and $O(\gamma ^n)$ respectively. I prove that the Raney numbers are a strict lower bound for enumerating sweep-covers on infinite $\Delta$-ary trees, $\Omega(\frac{(\Delta n)^n}{n!})$

An Introduction to Pursuit-evasion Differential Games

Pursuit and evasion conflicts represent challenging problems with important applications in aerospace and robotics. In pursuit-evasion problems, synthesis of intelligent actions must consider the adversary's potential strategies. Differential game theory provides an adequate framework to analyze possible outcomes of the conflict without assuming particular behaviors by the opponent. This article presents an organized introduction of pursuit-evasion differential games with an overview of recent advances in the area. First, a summary of the seminal work is outlined, highlighting important contributions. Next, more recent results are described by employing a classification based on the number of players: one-pursuer-one-evader, N-pursuers-one-evader, one-pursuer-M-evaders, and N-pursuer-M-evader games. In each scenario, a brief summary of the literature is presented. Finally, two representative pursuit-evasion differential games are studied in detail: the two-cutters and fugitive ship differential game and the active target defense differential game. These problems provide two important applications and, more importantly, they give great insight into the realization of cooperation between friendly agents in order to form a team and defeat the adversary.Comment: 18 pages, 7 figures, 2020 American Control Conference (tutorial paper