In this paper, relative controllability of impulsive multi-delay differential systems in finite dimensional space are studied. By introducing the impulsive multi-delay Gramian matrix, a necessary and sufficient condition, and the Gramian criteria, for the relative controllability of linear systems is given. Using Krasnoselskii’s fixed point theorem, a sufficient condition for controllability of semilinear systems is obtained. Numerically examples are given to illustrate our theoretically results.

中文翻译：

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脉冲多延迟微分系统的相对可控性

本文研究了有限维空间中脉冲多时滞微分系统的相对可控性。通过引入脉冲多时滞Gramian矩阵，给出了线性系统相对可控性的充分必要条件和Gramian准则。利用克拉斯诺塞尔斯基不动点定理，得到了半线性系统可控性的充分条件。给出了数值例子来说明我们的理论结果。