Abstract Necessary and sufficient conditions for the internal stability of formations whose dynamics are defined by linear differential equations have been obtained. The classes of admissible controls are specified as programmed controls for leaders and affine feedback controls depending on the object and leader states for followers. The conditions obtained are easy to verify and consist of (i) the stabilizability of a pair of matrices for the follower equations, (ii) the Hurwitz property and (iii) the coincidence of matrices for leaders in the multi-leader case, and (iv) the solvability of some linear equations and equality constraints on the vectors defining the desired relative leader–follower positions. Furthermore, the entire class of controls ensuring linear internal stability is described. By using the conditions obtained, it is shown that, in fact, only single-leader formations can possess internal stability. In the class of single-leader formations, a subclass of formations (whose graph is an input tree) is identified that are free of equality constraints, which are the main obstacle to the internal stability of multi-leader formations.

中文翻译：

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线性地层内部稳定的充要条件

摘要 得到了动力学由线性微分方程定义的地层内部稳定性的充要条件。根据对象和跟随者的领导状态，将允许控制的类别指定为领导者的程序控制和仿射反馈控制。获得的条件很容易验证，包括 (i) 跟随方程的一对矩阵的稳定性，(ii) Hurwitz 性质和 (iii) 在多领导者情况下领导者矩阵的重合，以及 ( iv) 一些线性方程的可解性和定义所需相对领导者-跟随者位置的向量的等式约束。此外，还描述了确保线性内部稳定性的整类控制。通过使用得到的条件，可以看出，事实上，只有单一领导的阵型才能具有内部稳定性。在单领导编队的类别中，确定了一个没有平等约束的编队子类（其图是输入树），这是多领导编队内部稳定性的主要障碍。