This paper investigates the problem of stabilization for multi-agent systems over finite fields via two types of impulsive control, that is, time-triggered impulsive control (TTIC) and state-triggered impulsive control (STIC). Inspired by the fact that finite-field networks (FFNs) cannot achieve stability at a nonzero state over the finite fields ${\mathbb{F}}_p,p>2$, two kinds impulse-controlled FFNs (ICFFNs) are introduced subject to time-triggering sequence and state-triggering set, respectively. In order to save time and control cost, the minimum-time control and minimum-triggering control of FFNs are both investigated. The time-optimal and triggering-optimal TTIC and STIC are successively designed. It reveals that designing such optimal TTIC for stabilizing ICFFNs can be transformed into finding the shortest path in the pretreated transition graph, and designing such optimal STIC for the global stabilization of ICFFNs is converted into finding the minimum spanning in-tree. Two examples are given to illustrate the control design procedures. Finally, we discuss the applications of the proposed control design methods on the synchronization problem and the switched FFNs including deterministic switched FFNs, probabilistic FFNs and Markovian jumping FFNs.

本文通过两种类型的脉冲控制，即时间触发脉冲控制 (TTIC) 和状态触发脉冲控制 (STIC)，研究了有限域上多智能体系统的镇定问题。受有限域网络 (FFN) 无法在有限域上的非零状态下实现稳定性这一事实的启发${\mathbb{F}}_{磷},磷>2$, 两种脉冲控制 FFNs (ICFFNs) 分别被引入时间触发序列和状态触发集。为了节省时间和控制成本，研究了FFN的最小时间控制和最小触发控制。先后设计了时间最优和触发最优的TTIC和STIC。它表明，设计这种用于稳定 ICFFNs 的最优 TTIC 可以转化为在预处理转移图中寻找最短路径，而设计这种用于 ICFFNs 全局稳定的最优 STIC 转化为寻找最小生成树。给出了两个例子来说明控制设计程序。最后，我们讨论了所提出的控制设计方法在同步问题和包括确定性切换 FFN 在内的切换 FFN 上的应用，