Kalman type rank condition for relative controllability of second order delayed linear difference equations

This paper explores the relative controllability of second-order delayed linear difference systems. Initially, we establish a Kalman-type rank condition as a fundamental criterion for assessing controllability. Following this, we introduce the discrete Gramian matrix and demonstrate its nonsingularity, supported by several auxiliary lemmas. Additionally, we derive a sufficient condition for relative controllability and construct an appropriate control function. To validate the theoretical results, a numerical example is presented that illustrates the practical implications and effectiveness of the proposed approach.